Calculus Examples

$\frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$

Step 1

Write $\frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ as a function.

$f (x) = \frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$

Step 2

Find the first derivative.

Tap for more steps...

Step 2.1

Find the first derivative.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $\frac{1}{5} x^{5} + \frac{7}{2} x^{4} + \frac{71}{3} x^{3} + 77 x^{2} + 120 x$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$ .

$\frac{d}{d x} [\frac{1}{5} x^{5}] + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{1}{5} x^{5}]$ .

Tap for more steps...

Step 2.1.2.1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5} x^{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x^{5}]$ .

$\frac{1}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{5} (5 x^{4}) + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.2.3

Combine $5$ and $\frac{1}{5}$ .

$\frac{5}{5} x^{4} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.2.4

Combine $\frac{5}{5}$ and $x^{4}$ .

$\frac{5 x^{4}}{5} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.2.5

Cancel the common factor of $5$ .

Tap for more steps...

Step 2.1.2.5.1

Cancel the common factor.

$\frac{5 x^{4}}{5} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.2.5.2

Divide $x^{4}$ by $1$ .

$x^{4} + \frac{d}{d x} [\frac{7}{2} x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3

Evaluate $\frac{d}{d x} [\frac{7}{2} x^{4}]$ .

Tap for more steps...

Step 2.1.3.1

Since $\frac{7}{2}$ is constant with respect to $x$ , the derivative of $\frac{7}{2} x^{4}$ with respect to $x$ is $\frac{7}{2} \frac{d}{d x} [x^{4}]$ .

$x^{4} + \frac{7}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$x^{4} + \frac{7}{2} (4 x^{3}) + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3.3

Combine $4$ and $\frac{7}{2}$ .

$x^{4} + \frac{4 \cdot 7}{2} x^{3} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3.4

Multiply $4$ by $7$ .

$x^{4} + \frac{28}{2} x^{3} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3.5

Combine $\frac{28}{2}$ and $x^{3}$ .

$x^{4} + \frac{28 x^{3}}{2} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3.6

Cancel the common factor of $28$ and $2$ .

Tap for more steps...

Step 2.1.3.6.1

Factor $2$ out of $28 x^{3}$ .

$x^{4} + \frac{2 (14 x^{3})}{2} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3.6.2

Cancel the common factors.

Tap for more steps...

Step 2.1.3.6.2.1

Factor $2$ out of $2$ .

$x^{4} + \frac{2 (14 x^{3})}{2 (1)} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3.6.2.2

Cancel the common factor.

$x^{4} + \frac{2 (14 x^{3})}{2 \cdot 1} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3.6.2.3

Rewrite the expression.

$x^{4} + \frac{14 x^{3}}{1} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.3.6.2.4

Divide $14 x^{3}$ by $1$ .

$x^{4} + 14 x^{3} + \frac{d}{d x} [\frac{71}{3} x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4

Evaluate $\frac{d}{d x} [\frac{71}{3} x^{3}]$ .

Tap for more steps...

Step 2.1.4.1

Since $\frac{71}{3}$ is constant with respect to $x$ , the derivative of $\frac{71}{3} x^{3}$ with respect to $x$ is $\frac{71}{3} \frac{d}{d x} [x^{3}]$ .

$x^{4} + 14 x^{3} + \frac{71}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{4} + 14 x^{3} + \frac{71}{3} (3 x^{2}) + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4.3

Combine $3$ and $\frac{71}{3}$ .

$x^{4} + 14 x^{3} + \frac{3 \cdot 71}{3} x^{2} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4.4

Multiply $3$ by $71$ .

$x^{4} + 14 x^{3} + \frac{213}{3} x^{2} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4.5

Combine $\frac{213}{3}$ and $x^{2}$ .

$x^{4} + 14 x^{3} + \frac{213 x^{2}}{3} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4.6

Cancel the common factor of $213$ and $3$ .

Tap for more steps...

Step 2.1.4.6.1

Factor $3$ out of $213 x^{2}$ .

$x^{4} + 14 x^{3} + \frac{3 (71 x^{2})}{3} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4.6.2

Cancel the common factors.

Tap for more steps...

Step 2.1.4.6.2.1

Factor $3$ out of $3$ .

$x^{4} + 14 x^{3} + \frac{3 (71 x^{2})}{3 (1)} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4.6.2.2

Cancel the common factor.

$x^{4} + 14 x^{3} + \frac{3 (71 x^{2})}{3 \cdot 1} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4.6.2.3

Rewrite the expression.

$x^{4} + 14 x^{3} + \frac{71 x^{2}}{1} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.4.6.2.4

Divide $71 x^{2}$ by $1$ .

$x^{4} + 14 x^{3} + 71 x^{2} + \frac{d}{d x} [77 x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.5

Evaluate $\frac{d}{d x} [77 x^{2}]$ .

Tap for more steps...

Step 2.1.5.1

Since $77$ is constant with respect to $x$ , the derivative of $77 x^{2}$ with respect to $x$ is $77 \frac{d}{d x} [x^{2}]$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 77 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [120 x]$

Step 2.1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 77 (2 x) + \frac{d}{d x} [120 x]$

Step 2.1.5.3

Multiply $2$ by $77$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + \frac{d}{d x} [120 x]$

Step 2.1.6

Evaluate $\frac{d}{d x} [120 x]$ .

Tap for more steps...

Step 2.1.6.1

Since $120$ is constant with respect to $x$ , the derivative of $120 x$ with respect to $x$ is $120 \frac{d}{d x} [x]$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 \frac{d}{d x} [x]$

Step 2.1.6.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 \cdot 1$

Step 2.1.6.3

Multiply $120$ by $1$ .

$f' (x) = x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$

Step 3

Set the first derivative equal to $0$ then solve the equation $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 = 0$ .

Tap for more steps...

Step 3.1

Set the first derivative equal to $0$ .

$x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120 = 0$

Step 3.2

Factor the left side of the equation.

Tap for more steps...

Step 3.2.1

Factor $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$ using the rational roots test.

Tap for more steps...

Step 3.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 120, \pm 2, \pm 60, \pm 3, \pm 40, \pm 4, \pm 30, \pm 5, \pm 24, \pm 6, \pm 20, \pm 8, \pm 15, \pm 10, \pm 12$

$q = \pm 1$

Step 3.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 120, \pm 2, \pm 60, \pm 3, \pm 40, \pm 4, \pm 30, \pm 5, \pm 24, \pm 6, \pm 20, \pm 8, \pm 15, \pm 10, \pm 12$

Step 3.2.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

Tap for more steps...

Step 3.2.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{4} + 14 {(- 2)}^{3} + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 3.2.1.3.2

Raise $- 2$ to the power of $4$ .

$16 + 14 {(- 2)}^{3} + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 3.2.1.3.3

Raise $- 2$ to the power of $3$ .

$16 + 14 \cdot - 8 + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 3.2.1.3.4

Multiply $14$ by $- 8$ .

$16 - 112 + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 3.2.1.3.5

Subtract $112$ from $16$ .

$- 96 + 71 {(- 2)}^{2} + 154 \cdot - 2 + 120$

Step 3.2.1.3.6

Raise $- 2$ to the power of $2$ .

$- 96 + 71 \cdot 4 + 154 \cdot - 2 + 120$

Step 3.2.1.3.7

Multiply $71$ by $4$ .

$- 96 + 284 + 154 \cdot - 2 + 120$

Step 3.2.1.3.8

Add $- 96$ and $284$ .

$188 + 154 \cdot - 2 + 120$

Step 3.2.1.3.9

Multiply $154$ by $- 2$ .

$188 - 308 + 120$

Step 3.2.1.3.10

Subtract $308$ from $188$ .

$- 120 + 120$

Step 3.2.1.3.11

Add $- 120$ and $120$ .

$0$

Step 3.2.1.4

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120}{x + 2}$

Step 3.2.1.5

Divide $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$ by $x + 2$ .

Tap for more steps...

Step 3.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

Step 3.2.1.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

Step 3.2.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

Step 3.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 2 x^{3}$

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

Step 3.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

Step 3.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

Step 3.2.1.5.7

Divide the highest order term in the dividend $12 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

Step 3.2.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

Step 3.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{3} + 24 x^{2}$

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

Step 3.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

Step 3.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$12 x^{2}$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

Step 3.2.1.5.12

Divide the highest order term in the dividend $47 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

Step 3.2.1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

Step 3.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $47 x^{2} + 94 x$

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

Step 3.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

Step 3.2.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{3}$

$12 x^{2}$

$47 x$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

Step 3.2.1.5.17

Divide the highest order term in the dividend $60 x$ by the highest order term in divisor $x$ .

$x^{3}$

$12 x^{2}$

$47 x$

$60$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

Step 3.2.1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$12 x^{2}$

$47 x$

$60$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

$60 x$

$120$

Step 3.2.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $60 x + 120$

$x^{3}$

$12 x^{2}$

$47 x$

$60$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

$60 x$

$120$

Step 3.2.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{3}$

$12 x^{2}$

$47 x$

$60$

$x$

$2$

$x^{4}$

$14 x^{3}$

$71 x^{2}$

$154 x$

$120$

$x^{4}$

$2 x^{3}$

$12 x^{3}$

$71 x^{2}$

$12 x^{3}$

$24 x^{2}$

$47 x^{2}$

$154 x$

$47 x^{2}$

$94 x$

$60 x$

$120$

$60 x$

$120$

$0$

Step 3.2.1.5.21

Since the remander is $0$ , the final answer is the quotient.

$x^{3} + 12 x^{2} + 47 x + 60$

Step 3.2.1.6

Write $x^{4} + 14 x^{3} + 71 x^{2} + 154 x + 120$ as a set of factors.

$(x + 2) (x^{3} + 12 x^{2} + 47 x + 60) = 0$

Step 3.2.2

Factor $x^{3} + 12 x^{2} + 47 x + 60$ using the rational roots test.

Tap for more steps...

Step 3.2.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10$

$q = \pm 1$

Step 3.2.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10$

Step 3.2.2.3

Substitute $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

Tap for more steps...

Step 3.2.2.3.1

Substitute $- 3$ into the polynomial.

${(- 3)}^{3} + 12 {(- 3)}^{2} + 47 \cdot - 3 + 60$

Step 3.2.2.3.2

Raise $- 3$ to the power of $3$ .

$- 27 + 12 {(- 3)}^{2} + 47 \cdot - 3 + 60$

Step 3.2.2.3.3

Raise $- 3$ to the power of $2$ .

$- 27 + 12 \cdot 9 + 47 \cdot - 3 + 60$

Step 3.2.2.3.4

Multiply $12$ by $9$ .

$- 27 + 108 + 47 \cdot - 3 + 60$

Step 3.2.2.3.5

Add $- 27$ and $108$ .

$81 + 47 \cdot - 3 + 60$

Step 3.2.2.3.6

Multiply $47$ by $- 3$ .

$81 - 141 + 60$

Step 3.2.2.3.7

Subtract $141$ from $81$ .

$- 60 + 60$

Step 3.2.2.3.8

Add $- 60$ and $60$ .

$0$

Step 3.2.2.4

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 12 x^{2} + 47 x + 60}{x + 3}$

Step 3.2.2.5

Divide $x^{3} + 12 x^{2} + 47 x + 60$ by $x + 3$ .

Tap for more steps...

Step 3.2.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$3$

$x^{3}$

$12 x^{2}$

$47 x$

$60$

Step 3.2.2.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$

Step 3.2.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			+	$x^{3}$	+	$3 x^{2}$

Step 3.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 3 x^{2}$

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$

Step 3.2.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$

Step 3.2.2.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$

Step 3.2.2.5.7

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$

Step 3.2.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					+	$9 x^{2}$	+	$27 x$

Step 3.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} + 27 x$

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$

Step 3.2.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$

Step 3.2.2.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$9 x$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$

Step 3.2.2.5.12

Divide the highest order term in the dividend $20 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$9 x$	+	$20$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$

Step 3.2.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$9 x$	+	$20$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$
							+	$20 x$	+	$60$

Step 3.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $20 x + 60$

				$x^{2}$	+	$9 x$	+	$20$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$
							-	$20 x$	-	$60$

Step 3.2.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$9 x$	+	$20$
$x$	+	$3$		$x^{3}$	+	$12 x^{2}$	+	$47 x$	+	$60$
			-	$x^{3}$	-	$3 x^{2}$
					+	$9 x^{2}$	+	$47 x$
					-	$9 x^{2}$	-	$27 x$
							+	$20 x$	+	$60$
							-	$20 x$	-	$60$
										$0$

Step 3.2.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 9 x + 20$

Step 3.2.2.6

Write $x^{3} + 12 x^{2} + 47 x + 60$ as a set of factors.

$(x + 2) ((x + 3) (x^{2} + 9 x + 20)) = 0$

Step 3.2.3

Factor $x^{2} + 9 x + 20$ using the AC method.

Tap for more steps...

Step 3.2.3.1

Factor $x^{2} + 9 x + 20$ using the AC method.

Tap for more steps...

Step 3.2.3.1.1

Factor $x^{2} + 9 x + 20$ using the AC method.

Tap for more steps...

Step 3.2.3.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $20$ and whose sum is $9$ .

$4, 5$

Step 3.2.3.1.1.2

Write the factored form using these integers.

$(x + 2) ((x + 3) ((x + 4) (x + 5))) = 0$

Step 3.2.3.1.2

Remove unnecessary parentheses.

$(x + 2) ((x + 3) (x + 4) (x + 5)) = 0$

Step 3.2.3.2

Remove unnecessary parentheses.

$(x + 2) (x + 3) (x + 4) (x + 5) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x + 3 = 0$

$x + 4 = 0$

$x + 5 = 0$

Step 3.4

Set $x + 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.4.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.5

Set $x + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.6

Set $x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.6.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 3.6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.7

Set $x + 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.7.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 3.7.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 3.8

The final solution is all the values that make $(x + 2) (x + 3) (x + 4) (x + 5) = 0$ true.

$x = - 2, - 3, - 4, - 5$

Step 4

The values which make the derivative equal to $0$ are $- 2, - 3, - 4, - 5$ .

$- 2, - 3, - 4, - 5$

Step 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, - 5) \cup (- 5, - 4) \cup (- 4, - 3) \cup (- 3, - 2) \cup (- 2, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 5)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 6$ in the expression.

$f' (- 6) = {(- 6)}^{4} + 14 {(- 6)}^{3} + 71 {(- 6)}^{2} + 154 (- 6) + 120$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Raise $- 6$ to the power of $4$ .

$f' (- 6) = 1296 + 14 {(- 6)}^{3} + 71 {(- 6)}^{2} + 154 (- 6) + 120$

Step 6.2.1.2

Raise $- 6$ to the power of $3$ .

$f' (- 6) = 1296 + 14 \cdot - 216 + 71 {(- 6)}^{2} + 154 (- 6) + 120$

Step 6.2.1.3

Multiply $14$ by $- 216$ .

$f' (- 6) = 1296 - 3024 + 71 {(- 6)}^{2} + 154 (- 6) + 120$

Step 6.2.1.4

Raise $- 6$ to the power of $2$ .

$f' (- 6) = 1296 - 3024 + 71 \cdot 36 + 154 (- 6) + 120$

Step 6.2.1.5

Multiply $71$ by $36$ .

$f' (- 6) = 1296 - 3024 + 2556 + 154 (- 6) + 120$

Step 6.2.1.6

Multiply $154$ by $- 6$ .

$f' (- 6) = 1296 - 3024 + 2556 - 924 + 120$

Step 6.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 6.2.2.1

Subtract $3024$ from $1296$ .

$f' (- 6) = - 1728 + 2556 - 924 + 120$

Step 6.2.2.2

Add $- 1728$ and $2556$ .

$f' (- 6) = 828 - 924 + 120$

Step 6.2.2.3

Subtract $924$ from $828$ .

$f' (- 6) = - 96 + 120$

Step 6.2.2.4

Add $- 96$ and $120$ .

$f' (- 6) = 24$

Step 6.2.3

The final answer is $24$ .

$24$

Step 6.3

At $x = - 6$ the derivative is $24$ . Since this is positive, the function is increasing on $(- \infty, - 5)$ .

Increasing on $(- \infty, - 5)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(- 5, - 4)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $- \frac{9}{2}$ in the expression.

$f' (- \frac{9}{2}) = {(- \frac{9}{2})}^{4} + 14 {(- \frac{9}{2})}^{3} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 7.2.1.1.1

Apply the product rule to $- \frac{9}{2}$ .

$f' (- \frac{9}{2}) = {(- 1)}^{4} {(\frac{9}{2})}^{4} + 14 {(- \frac{9}{2})}^{3} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.1.2

Apply the product rule to $\frac{9}{2}$ .

$f' (- \frac{9}{2}) = {(- 1)}^{4} (\frac{9^{4}}{2^{4}}) + 14 {(- \frac{9}{2})}^{3} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.2

Raise $- 1$ to the power of $4$ .

$f' (- \frac{9}{2}) = 1 (\frac{9^{4}}{2^{4}}) + 14 {(- \frac{9}{2})}^{3} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.3

Multiply $\frac{9^{4}}{2^{4}}$ by $1$ .

$f' (- \frac{9}{2}) = \frac{9^{4}}{2^{4}} + 14 {(- \frac{9}{2})}^{3} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.4

Raise $9$ to the power of $4$ .

$f' (- \frac{9}{2}) = \frac{6561}{2^{4}} + 14 {(- \frac{9}{2})}^{3} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.5

Raise $2$ to the power of $4$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + 14 {(- \frac{9}{2})}^{3} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 7.2.1.6.1

Apply the product rule to $- \frac{9}{2}$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + 14 ({(- 1)}^{3} {(\frac{9}{2})}^{3}) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.6.2

Apply the product rule to $\frac{9}{2}$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + 14 ({(- 1)}^{3} (\frac{9^{3}}{2^{3}})) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.7

Raise $- 1$ to the power of $3$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + 14 (- \frac{9^{3}}{2^{3}}) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.8

Raise $9$ to the power of $3$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + 14 (- \frac{729}{2^{3}}) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.9

Raise $2$ to the power of $3$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + 14 (- \frac{729}{8}) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.10

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.1.10.1

Move the leading negative in $- \frac{729}{8}$ into the numerator.

$f' (- \frac{9}{2}) = \frac{6561}{16} + 14 (\frac{- 729}{8}) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.10.2

Factor $2$ out of $14$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + 2 (7) (\frac{- 729}{8}) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.10.3

Factor $2$ out of $8$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + 2 \cdot (7 (\frac{- 729}{2 \cdot 4})) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.10.4

Cancel the common factor.

$f' (- \frac{9}{2}) = \frac{6561}{16} + 2 \cdot (7 (\frac{- 729}{2 \cdot 4})) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.10.5

Rewrite the expression.

$f' (- \frac{9}{2}) = \frac{6561}{16} + 7 (\frac{- 729}{4}) + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.11

Combine $7$ and $\frac{- 729}{4}$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + \frac{7 \cdot - 729}{4} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.12

Multiply $7$ by $- 729$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} + \frac{- 5103}{4} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.13

Move the negative in front of the fraction.

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + 71 {(- \frac{9}{2})}^{2} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.14

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 7.2.1.14.1

Apply the product rule to $- \frac{9}{2}$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + 71 ({(- 1)}^{2} {(\frac{9}{2})}^{2}) + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.14.2

Apply the product rule to $\frac{9}{2}$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + 71 ({(- 1)}^{2} (\frac{9^{2}}{2^{2}})) + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.15

Raise $- 1$ to the power of $2$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + 71 (1 (\frac{9^{2}}{2^{2}})) + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.16

Multiply $\frac{9^{2}}{2^{2}}$ by $1$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + 71 (\frac{9^{2}}{2^{2}}) + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.17

Raise $9$ to the power of $2$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + 71 (\frac{81}{2^{2}}) + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.18

Raise $2$ to the power of $2$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + 71 (\frac{81}{4}) + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.19

Multiply $71 (\frac{81}{4})$ .

Tap for more steps...

Step 7.2.1.19.1

Combine $71$ and $\frac{81}{4}$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + \frac{71 \cdot 81}{4} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.19.2

Multiply $71$ by $81$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + \frac{5751}{4} + 154 (- \frac{9}{2}) + 120$

Step 7.2.1.20

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.1.20.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + \frac{5751}{4} + 154 (\frac{- 9}{2}) + 120$

Step 7.2.1.20.2

Factor $2$ out of $154$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + \frac{5751}{4} + 2 (77) (\frac{- 9}{2}) + 120$

Step 7.2.1.20.3

Cancel the common factor.

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + \frac{5751}{4} + 2 \cdot (77 (\frac{- 9}{2})) + 120$

Step 7.2.1.20.4

Rewrite the expression.

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + \frac{5751}{4} + 77 \cdot - 9 + 120$

Step 7.2.1.21

Multiply $77$ by $- 9$ .

$f' (- \frac{9}{2}) = \frac{6561}{16} - \frac{5103}{4} + \frac{5751}{4} - 693 + 120$

Step 7.2.2

Combine fractions.

Tap for more steps...

Step 7.2.2.1

Combine the numerators over the common denominator.

$f' (- \frac{9}{2}) = - 693 + 120 + \frac{6561}{16} + \frac{- 5103 + 5751}{4}$

Step 7.2.2.2

Add $- 5103$ and $5751$ .

$f' (- \frac{9}{2}) = - 693 + 120 + \frac{6561}{16} + \frac{648}{4}$

Step 7.2.3

Find the common denominator.

Tap for more steps...

Step 7.2.3.1

Write $- 693$ as a fraction with denominator $1$ .

$f' (- \frac{9}{2}) = \frac{- 693}{1} + 120 + \frac{6561}{16} + \frac{648}{4}$

Step 7.2.3.2

Multiply $\frac{- 693}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{9}{2}) = \frac{- 693}{1} \cdot \frac{16}{16} + 120 + \frac{6561}{16} + \frac{648}{4}$

Step 7.2.3.3

Multiply $\frac{- 693}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{9}{2}) = \frac{- 693 \cdot 16}{16} + 120 + \frac{6561}{16} + \frac{648}{4}$

Step 7.2.3.4

Write $120$ as a fraction with denominator $1$ .

$f' (- \frac{9}{2}) = \frac{- 693 \cdot 16}{16} + \frac{120}{1} + \frac{6561}{16} + \frac{648}{4}$

Step 7.2.3.5

Multiply $\frac{120}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{9}{2}) = \frac{- 693 \cdot 16}{16} + \frac{120}{1} \cdot \frac{16}{16} + \frac{6561}{16} + \frac{648}{4}$

Step 7.2.3.6

Multiply $\frac{120}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{9}{2}) = \frac{- 693 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{6561}{16} + \frac{648}{4}$

Step 7.2.3.7

Multiply $\frac{648}{4}$ by $\frac{4}{4}$ .

$f' (- \frac{9}{2}) = \frac{- 693 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{6561}{16} + \frac{648}{4} \cdot \frac{4}{4}$

Step 7.2.3.8

Multiply $\frac{648}{4}$ by $\frac{4}{4}$ .

$f' (- \frac{9}{2}) = \frac{- 693 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{6561}{16} + \frac{648 \cdot 4}{4 \cdot 4}$

Step 7.2.3.9

Multiply $4$ by $4$ .

$f' (- \frac{9}{2}) = \frac{- 693 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{6561}{16} + \frac{648 \cdot 4}{16}$

Step 7.2.4

Combine the numerators over the common denominator.

$f' (- \frac{9}{2}) = \frac{- 693 \cdot 16 + 120 \cdot 16 + 6561 + 648 \cdot 4}{16}$

Step 7.2.5

Simplify each term.

Tap for more steps...

Step 7.2.5.1

Multiply $- 693$ by $16$ .

$f' (- \frac{9}{2}) = \frac{- 11088 + 120 \cdot 16 + 6561 + 648 \cdot 4}{16}$

Step 7.2.5.2

Multiply $120$ by $16$ .

$f' (- \frac{9}{2}) = \frac{- 11088 + 1920 + 6561 + 648 \cdot 4}{16}$

Step 7.2.5.3

Multiply $648$ by $4$ .

$f' (- \frac{9}{2}) = \frac{- 11088 + 1920 + 6561 + 2592}{16}$

Step 7.2.6

Simplify the expression.

Tap for more steps...

Step 7.2.6.1

Add $- 11088$ and $1920$ .

$f' (- \frac{9}{2}) = \frac{- 9168 + 6561 + 2592}{16}$

Step 7.2.6.2

Add $- 9168$ and $6561$ .

$f' (- \frac{9}{2}) = \frac{- 2607 + 2592}{16}$

Step 7.2.6.3

Add $- 2607$ and $2592$ .

$f' (- \frac{9}{2}) = \frac{- 15}{16}$

Step 7.2.6.4

Move the negative in front of the fraction.

$f' (- \frac{9}{2}) = - \frac{15}{16}$

Step 7.2.7

The final answer is $- \frac{15}{16}$ .

$- \frac{15}{16}$

Step 7.3

At $x = - \frac{9}{2}$ the derivative is $- \frac{15}{16}$ . Since this is negative, the function is decreasing on $(- 5, - 4)$ .

Decreasing on $(- 5, - 4)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(- 4, - 3)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $- \frac{7}{2}$ in the expression.

$f' (- \frac{7}{2}) = {(- \frac{7}{2})}^{4} + 14 {(- \frac{7}{2})}^{3} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Simplify each term.

Tap for more steps...

Step 8.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 8.2.1.1.1

Apply the product rule to $- \frac{7}{2}$ .

$f' (- \frac{7}{2}) = {(- 1)}^{4} {(\frac{7}{2})}^{4} + 14 {(- \frac{7}{2})}^{3} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.1.2

Apply the product rule to $\frac{7}{2}$ .

$f' (- \frac{7}{2}) = {(- 1)}^{4} (\frac{7^{4}}{2^{4}}) + 14 {(- \frac{7}{2})}^{3} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.2

Raise $- 1$ to the power of $4$ .

$f' (- \frac{7}{2}) = 1 (\frac{7^{4}}{2^{4}}) + 14 {(- \frac{7}{2})}^{3} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.3

Multiply $\frac{7^{4}}{2^{4}}$ by $1$ .

$f' (- \frac{7}{2}) = \frac{7^{4}}{2^{4}} + 14 {(- \frac{7}{2})}^{3} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.4

Raise $7$ to the power of $4$ .

$f' (- \frac{7}{2}) = \frac{2401}{2^{4}} + 14 {(- \frac{7}{2})}^{3} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.5

Raise $2$ to the power of $4$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + 14 {(- \frac{7}{2})}^{3} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 8.2.1.6.1

Apply the product rule to $- \frac{7}{2}$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + 14 ({(- 1)}^{3} {(\frac{7}{2})}^{3}) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.6.2

Apply the product rule to $\frac{7}{2}$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + 14 ({(- 1)}^{3} (\frac{7^{3}}{2^{3}})) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.7

Raise $- 1$ to the power of $3$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + 14 (- \frac{7^{3}}{2^{3}}) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.8

Raise $7$ to the power of $3$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + 14 (- \frac{343}{2^{3}}) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.9

Raise $2$ to the power of $3$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + 14 (- \frac{343}{8}) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.10

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.2.1.10.1

Move the leading negative in $- \frac{343}{8}$ into the numerator.

$f' (- \frac{7}{2}) = \frac{2401}{16} + 14 (\frac{- 343}{8}) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.10.2

Factor $2$ out of $14$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + 2 (7) (\frac{- 343}{8}) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.10.3

Factor $2$ out of $8$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + 2 \cdot (7 (\frac{- 343}{2 \cdot 4})) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.10.4

Cancel the common factor.

$f' (- \frac{7}{2}) = \frac{2401}{16} + 2 \cdot (7 (\frac{- 343}{2 \cdot 4})) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.10.5

Rewrite the expression.

$f' (- \frac{7}{2}) = \frac{2401}{16} + 7 (\frac{- 343}{4}) + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.11

Combine $7$ and $\frac{- 343}{4}$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + \frac{7 \cdot - 343}{4} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.12

Multiply $7$ by $- 343$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} + \frac{- 2401}{4} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.13

Move the negative in front of the fraction.

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + 71 {(- \frac{7}{2})}^{2} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.14

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 8.2.1.14.1

Apply the product rule to $- \frac{7}{2}$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + 71 ({(- 1)}^{2} {(\frac{7}{2})}^{2}) + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.14.2

Apply the product rule to $\frac{7}{2}$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + 71 ({(- 1)}^{2} (\frac{7^{2}}{2^{2}})) + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.15

Raise $- 1$ to the power of $2$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + 71 (1 (\frac{7^{2}}{2^{2}})) + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.16

Multiply $\frac{7^{2}}{2^{2}}$ by $1$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + 71 (\frac{7^{2}}{2^{2}}) + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.17

Raise $7$ to the power of $2$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + 71 (\frac{49}{2^{2}}) + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.18

Raise $2$ to the power of $2$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + 71 (\frac{49}{4}) + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.19

Multiply $71 (\frac{49}{4})$ .

Tap for more steps...

Step 8.2.1.19.1

Combine $71$ and $\frac{49}{4}$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + \frac{71 \cdot 49}{4} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.19.2

Multiply $71$ by $49$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + \frac{3479}{4} + 154 (- \frac{7}{2}) + 120$

Step 8.2.1.20

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.2.1.20.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + \frac{3479}{4} + 154 (\frac{- 7}{2}) + 120$

Step 8.2.1.20.2

Factor $2$ out of $154$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + \frac{3479}{4} + 2 (77) (\frac{- 7}{2}) + 120$

Step 8.2.1.20.3

Cancel the common factor.

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + \frac{3479}{4} + 2 \cdot (77 (\frac{- 7}{2})) + 120$

Step 8.2.1.20.4

Rewrite the expression.

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + \frac{3479}{4} + 77 \cdot - 7 + 120$

Step 8.2.1.21

Multiply $77$ by $- 7$ .

$f' (- \frac{7}{2}) = \frac{2401}{16} - \frac{2401}{4} + \frac{3479}{4} - 539 + 120$

Step 8.2.2

Combine fractions.

Tap for more steps...

Step 8.2.2.1

Combine the numerators over the common denominator.

$f' (- \frac{7}{2}) = - 539 + 120 + \frac{2401}{16} + \frac{- 2401 + 3479}{4}$

Step 8.2.2.2

Add $- 2401$ and $3479$ .

$f' (- \frac{7}{2}) = - 539 + 120 + \frac{2401}{16} + \frac{1078}{4}$

Step 8.2.3

Find the common denominator.

Tap for more steps...

Step 8.2.3.1

Write $- 539$ as a fraction with denominator $1$ .

$f' (- \frac{7}{2}) = \frac{- 539}{1} + 120 + \frac{2401}{16} + \frac{1078}{4}$

Step 8.2.3.2

Multiply $\frac{- 539}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{7}{2}) = \frac{- 539}{1} \cdot \frac{16}{16} + 120 + \frac{2401}{16} + \frac{1078}{4}$

Step 8.2.3.3

Multiply $\frac{- 539}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{7}{2}) = \frac{- 539 \cdot 16}{16} + 120 + \frac{2401}{16} + \frac{1078}{4}$

Step 8.2.3.4

Write $120$ as a fraction with denominator $1$ .

$f' (- \frac{7}{2}) = \frac{- 539 \cdot 16}{16} + \frac{120}{1} + \frac{2401}{16} + \frac{1078}{4}$

Step 8.2.3.5

Multiply $\frac{120}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{7}{2}) = \frac{- 539 \cdot 16}{16} + \frac{120}{1} \cdot \frac{16}{16} + \frac{2401}{16} + \frac{1078}{4}$

Step 8.2.3.6

Multiply $\frac{120}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{7}{2}) = \frac{- 539 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{2401}{16} + \frac{1078}{4}$

Step 8.2.3.7

Multiply $\frac{1078}{4}$ by $\frac{4}{4}$ .

$f' (- \frac{7}{2}) = \frac{- 539 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{2401}{16} + \frac{1078}{4} \cdot \frac{4}{4}$

Step 8.2.3.8

Multiply $\frac{1078}{4}$ by $\frac{4}{4}$ .

$f' (- \frac{7}{2}) = \frac{- 539 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{2401}{16} + \frac{1078 \cdot 4}{4 \cdot 4}$

Step 8.2.3.9

Multiply $4$ by $4$ .

$f' (- \frac{7}{2}) = \frac{- 539 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{2401}{16} + \frac{1078 \cdot 4}{16}$

Step 8.2.4

Combine the numerators over the common denominator.

$f' (- \frac{7}{2}) = \frac{- 539 \cdot 16 + 120 \cdot 16 + 2401 + 1078 \cdot 4}{16}$

Step 8.2.5

Simplify each term.

Tap for more steps...

Step 8.2.5.1

Multiply $- 539$ by $16$ .

$f' (- \frac{7}{2}) = \frac{- 8624 + 120 \cdot 16 + 2401 + 1078 \cdot 4}{16}$

Step 8.2.5.2

Multiply $120$ by $16$ .

$f' (- \frac{7}{2}) = \frac{- 8624 + 1920 + 2401 + 1078 \cdot 4}{16}$

Step 8.2.5.3

Multiply $1078$ by $4$ .

$f' (- \frac{7}{2}) = \frac{- 8624 + 1920 + 2401 + 4312}{16}$

Step 8.2.6

Simplify by adding numbers.

Tap for more steps...

Step 8.2.6.1

Add $- 8624$ and $1920$ .

$f' (- \frac{7}{2}) = \frac{- 6704 + 2401 + 4312}{16}$

Step 8.2.6.2

Add $- 6704$ and $2401$ .

$f' (- \frac{7}{2}) = \frac{- 4303 + 4312}{16}$

Step 8.2.6.3

Add $- 4303$ and $4312$ .

$f' (- \frac{7}{2}) = \frac{9}{16}$

Step 8.2.7

The final answer is $\frac{9}{16}$ .

$\frac{9}{16}$

Step 8.3

At $x = - \frac{7}{2}$ the derivative is $\frac{9}{16}$ . Since this is positive, the function is increasing on $(- 4, - 3)$ .

Increasing on $(- 4, - 3)$ since $f' (x) > 0$

Step 9

Substitute a value from the interval $(- 3, - 2)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 9.1

Replace the variable $x$ with $- \frac{5}{2}$ in the expression.

$f' (- \frac{5}{2}) = {(- \frac{5}{2})}^{4} + 14 {(- \frac{5}{2})}^{3} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2

Simplify the result.

Tap for more steps...

Step 9.2.1

Simplify each term.

Tap for more steps...

Step 9.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 9.2.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$f' (- \frac{5}{2}) = {(- 1)}^{4} {(\frac{5}{2})}^{4} + 14 {(- \frac{5}{2})}^{3} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.1.2

Apply the product rule to $\frac{5}{2}$ .

$f' (- \frac{5}{2}) = {(- 1)}^{4} (\frac{5^{4}}{2^{4}}) + 14 {(- \frac{5}{2})}^{3} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.2

Raise $- 1$ to the power of $4$ .

$f' (- \frac{5}{2}) = 1 (\frac{5^{4}}{2^{4}}) + 14 {(- \frac{5}{2})}^{3} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.3

Multiply $\frac{5^{4}}{2^{4}}$ by $1$ .

$f' (- \frac{5}{2}) = \frac{5^{4}}{2^{4}} + 14 {(- \frac{5}{2})}^{3} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.4

Raise $5$ to the power of $4$ .

$f' (- \frac{5}{2}) = \frac{625}{2^{4}} + 14 {(- \frac{5}{2})}^{3} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.5

Raise $2$ to the power of $4$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + 14 {(- \frac{5}{2})}^{3} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 9.2.1.6.1

Apply the product rule to $- \frac{5}{2}$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + 14 ({(- 1)}^{3} {(\frac{5}{2})}^{3}) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.6.2

Apply the product rule to $\frac{5}{2}$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + 14 ({(- 1)}^{3} (\frac{5^{3}}{2^{3}})) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.7

Raise $- 1$ to the power of $3$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + 14 (- \frac{5^{3}}{2^{3}}) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.8

Raise $5$ to the power of $3$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + 14 (- \frac{125}{2^{3}}) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.9

Raise $2$ to the power of $3$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + 14 (- \frac{125}{8}) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.10

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.2.1.10.1

Move the leading negative in $- \frac{125}{8}$ into the numerator.

$f' (- \frac{5}{2}) = \frac{625}{16} + 14 (\frac{- 125}{8}) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.10.2

Factor $2$ out of $14$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + 2 (7) (\frac{- 125}{8}) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.10.3

Factor $2$ out of $8$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + 2 \cdot (7 (\frac{- 125}{2 \cdot 4})) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.10.4

Cancel the common factor.

$f' (- \frac{5}{2}) = \frac{625}{16} + 2 \cdot (7 (\frac{- 125}{2 \cdot 4})) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.10.5

Rewrite the expression.

$f' (- \frac{5}{2}) = \frac{625}{16} + 7 (\frac{- 125}{4}) + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.11

Combine $7$ and $\frac{- 125}{4}$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + \frac{7 \cdot - 125}{4} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.12

Multiply $7$ by $- 125$ .

$f' (- \frac{5}{2}) = \frac{625}{16} + \frac{- 875}{4} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.13

Move the negative in front of the fraction.

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + 71 {(- \frac{5}{2})}^{2} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.14

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 9.2.1.14.1

Apply the product rule to $- \frac{5}{2}$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + 71 ({(- 1)}^{2} {(\frac{5}{2})}^{2}) + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.14.2

Apply the product rule to $\frac{5}{2}$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + 71 ({(- 1)}^{2} (\frac{5^{2}}{2^{2}})) + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.15

Raise $- 1$ to the power of $2$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + 71 (1 (\frac{5^{2}}{2^{2}})) + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.16

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + 71 (\frac{5^{2}}{2^{2}}) + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.17

Raise $5$ to the power of $2$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + 71 (\frac{25}{2^{2}}) + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.18

Raise $2$ to the power of $2$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + 71 (\frac{25}{4}) + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.19

Multiply $71 (\frac{25}{4})$ .

Tap for more steps...

Step 9.2.1.19.1

Combine $71$ and $\frac{25}{4}$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + \frac{71 \cdot 25}{4} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.19.2

Multiply $71$ by $25$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + \frac{1775}{4} + 154 (- \frac{5}{2}) + 120$

Step 9.2.1.20

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.2.1.20.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + \frac{1775}{4} + 154 (\frac{- 5}{2}) + 120$

Step 9.2.1.20.2

Factor $2$ out of $154$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + \frac{1775}{4} + 2 (77) (\frac{- 5}{2}) + 120$

Step 9.2.1.20.3

Cancel the common factor.

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + \frac{1775}{4} + 2 \cdot (77 (\frac{- 5}{2})) + 120$

Step 9.2.1.20.4

Rewrite the expression.

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + \frac{1775}{4} + 77 \cdot - 5 + 120$

Step 9.2.1.21

Multiply $77$ by $- 5$ .

$f' (- \frac{5}{2}) = \frac{625}{16} - \frac{875}{4} + \frac{1775}{4} - 385 + 120$

Step 9.2.2

Combine fractions.

Tap for more steps...

Step 9.2.2.1

Combine the numerators over the common denominator.

$f' (- \frac{5}{2}) = - 385 + 120 + \frac{625}{16} + \frac{- 875 + 1775}{4}$

Step 9.2.2.2

Add $- 875$ and $1775$ .

$f' (- \frac{5}{2}) = - 385 + 120 + \frac{625}{16} + \frac{900}{4}$

Step 9.2.3

Find the common denominator.

Tap for more steps...

Step 9.2.3.1

Write $- 385$ as a fraction with denominator $1$ .

$f' (- \frac{5}{2}) = \frac{- 385}{1} + 120 + \frac{625}{16} + \frac{900}{4}$

Step 9.2.3.2

Multiply $\frac{- 385}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{5}{2}) = \frac{- 385}{1} \cdot \frac{16}{16} + 120 + \frac{625}{16} + \frac{900}{4}$

Step 9.2.3.3

Multiply $\frac{- 385}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{5}{2}) = \frac{- 385 \cdot 16}{16} + 120 + \frac{625}{16} + \frac{900}{4}$

Step 9.2.3.4

Write $120$ as a fraction with denominator $1$ .

$f' (- \frac{5}{2}) = \frac{- 385 \cdot 16}{16} + \frac{120}{1} + \frac{625}{16} + \frac{900}{4}$

Step 9.2.3.5

Multiply $\frac{120}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{5}{2}) = \frac{- 385 \cdot 16}{16} + \frac{120}{1} \cdot \frac{16}{16} + \frac{625}{16} + \frac{900}{4}$

Step 9.2.3.6

Multiply $\frac{120}{1}$ by $\frac{16}{16}$ .

$f' (- \frac{5}{2}) = \frac{- 385 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{625}{16} + \frac{900}{4}$

Step 9.2.3.7

Multiply $\frac{900}{4}$ by $\frac{4}{4}$ .

$f' (- \frac{5}{2}) = \frac{- 385 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{625}{16} + \frac{900}{4} \cdot \frac{4}{4}$

Step 9.2.3.8

Multiply $\frac{900}{4}$ by $\frac{4}{4}$ .

$f' (- \frac{5}{2}) = \frac{- 385 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{625}{16} + \frac{900 \cdot 4}{4 \cdot 4}$

Step 9.2.3.9

Multiply $4$ by $4$ .

$f' (- \frac{5}{2}) = \frac{- 385 \cdot 16}{16} + \frac{120 \cdot 16}{16} + \frac{625}{16} + \frac{900 \cdot 4}{16}$

Step 9.2.4

Combine the numerators over the common denominator.

$f' (- \frac{5}{2}) = \frac{- 385 \cdot 16 + 120 \cdot 16 + 625 + 900 \cdot 4}{16}$

Step 9.2.5

Simplify each term.

Tap for more steps...

Step 9.2.5.1

Multiply $- 385$ by $16$ .

$f' (- \frac{5}{2}) = \frac{- 6160 + 120 \cdot 16 + 625 + 900 \cdot 4}{16}$

Step 9.2.5.2

Multiply $120$ by $16$ .

$f' (- \frac{5}{2}) = \frac{- 6160 + 1920 + 625 + 900 \cdot 4}{16}$

Step 9.2.5.3

Multiply $900$ by $4$ .

$f' (- \frac{5}{2}) = \frac{- 6160 + 1920 + 625 + 3600}{16}$

Step 9.2.6

Simplify the expression.

Tap for more steps...

Step 9.2.6.1

Add $- 6160$ and $1920$ .

$f' (- \frac{5}{2}) = \frac{- 4240 + 625 + 3600}{16}$

Step 9.2.6.2

Add $- 4240$ and $625$ .

$f' (- \frac{5}{2}) = \frac{- 3615 + 3600}{16}$

Step 9.2.6.3

Add $- 3615$ and $3600$ .

$f' (- \frac{5}{2}) = \frac{- 15}{16}$

Step 9.2.6.4

Move the negative in front of the fraction.

$f' (- \frac{5}{2}) = - \frac{15}{16}$

Step 9.2.7

The final answer is $- \frac{15}{16}$ .

$- \frac{15}{16}$

Step 9.3

At $x = - \frac{5}{2}$ the derivative is $- \frac{15}{16}$ . Since this is negative, the function is decreasing on $(- 3, - 2)$ .

Decreasing on $(- 3, - 2)$ since $f' (x) < 0$

Step 10

Substitute a value from the interval $(- 2, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 10.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = {(- 1)}^{4} + 14 {(- 1)}^{3} + 71 {(- 1)}^{2} + 154 (- 1) + 120$

Step 10.2

Simplify the result.

Tap for more steps...

Step 10.2.1

Simplify each term.

Tap for more steps...

Step 10.2.1.1

Raise $- 1$ to the power of $4$ .

$f' (- 1) = 1 + 14 {(- 1)}^{3} + 71 {(- 1)}^{2} + 154 (- 1) + 120$

Step 10.2.1.2

Raise $- 1$ to the power of $3$ .

$f' (- 1) = 1 + 14 \cdot - 1 + 71 {(- 1)}^{2} + 154 (- 1) + 120$

Step 10.2.1.3

Multiply $14$ by $- 1$ .

$f' (- 1) = 1 - 14 + 71 {(- 1)}^{2} + 154 (- 1) + 120$

Step 10.2.1.4

Raise $- 1$ to the power of $2$ .

$f' (- 1) = 1 - 14 + 71 \cdot 1 + 154 (- 1) + 120$

Step 10.2.1.5

Multiply $71$ by $1$ .

$f' (- 1) = 1 - 14 + 71 + 154 (- 1) + 120$

Step 10.2.1.6

Multiply $154$ by $- 1$ .

$f' (- 1) = 1 - 14 + 71 - 154 + 120$

Step 10.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 10.2.2.1

Subtract $14$ from $1$ .

$f' (- 1) = - 13 + 71 - 154 + 120$

Step 10.2.2.2

Add $- 13$ and $71$ .

$f' (- 1) = 58 - 154 + 120$

Step 10.2.2.3

Subtract $154$ from $58$ .

$f' (- 1) = - 96 + 120$

Step 10.2.2.4

Add $- 96$ and $120$ .

$f' (- 1) = 24$

Step 10.2.3

The final answer is $24$ .

$24$

Step 10.3

At $x = - 1$ the derivative is $24$ . Since this is positive, the function is increasing on $(- 2, \infty)$ .

Increasing on $(- 2, \infty)$ since $f' (x) > 0$

Step 11

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - 5), (- 4, - 3), (- 2, \infty)$

Decreasing on: $(- 5, - 4), (- 3, - 2)$

Step 12