Algebra Examples

$f (x) = 6 x^{4} - 21 x^{3} - 4 x^{2} + 24 x - 35$

Step 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.

$p = \pm 1, \pm 5, \pm 7, \pm 35$

$q = \pm 1, \pm 2, \pm 3, \pm 6$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{3}, \pm \frac{5}{6}, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{3}, \pm \frac{7}{6}, \pm 35, \pm \frac{35}{2}, \pm \frac{35}{3}, \pm \frac{35}{6}$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$6 {(\frac{7}{2})}^{4} - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = \frac{7}{2}$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$6 \frac{7^{4}}{2^{4}} - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.2

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

$6 \frac{2401}{2^{4}} - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.3

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

$6 (\frac{2401}{16}) - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Factor $2$ out of $6$ .

$2 (3) \frac{2401}{16} - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.4.2

Factor $2$ out of $16$ .

$2 \cdot 3 \frac{2401}{2 \cdot 8} - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.4.3

Cancel the common factor.

$2 \cdot 3 \frac{2401}{2 \cdot 8} - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.4.4

Rewrite the expression.

$3 (\frac{2401}{8}) - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.5

Combine $3$ and $\frac{2401}{8}$ .

$\frac{3 \cdot 2401}{8} - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.6

Multiply $3$ by $2401$ .

$\frac{7203}{8} - 21 {(\frac{7}{2})}^{3} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.7

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

$\frac{7203}{8} - 21 \frac{7^{3}}{2^{3}} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.8

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

$\frac{7203}{8} - 21 \frac{343}{2^{3}} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.9

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

$\frac{7203}{8} - 21 (\frac{343}{8}) - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.10

Multiply $- 21 (\frac{343}{8})$ .

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Step 4.1.10.1

Combine $- 21$ and $\frac{343}{8}$ .

$\frac{7203}{8} + \frac{- 21 \cdot 343}{8} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.10.2

Multiply $- 21$ by $343$ .

$\frac{7203}{8} + \frac{- 7203}{8} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.11

Move the negative in front of the fraction.

$\frac{7203}{8} - \frac{7203}{8} - 4 {(\frac{7}{2})}^{2} + 24 (\frac{7}{2}) - 35$

Step 4.1.12

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

$\frac{7203}{8} - \frac{7203}{8} - 4 \frac{7^{2}}{2^{2}} + 24 (\frac{7}{2}) - 35$

Step 4.1.13

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

$\frac{7203}{8} - \frac{7203}{8} - 4 \frac{49}{2^{2}} + 24 (\frac{7}{2}) - 35$

Step 4.1.14

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

$\frac{7203}{8} - \frac{7203}{8} - 4 (\frac{49}{4}) + 24 (\frac{7}{2}) - 35$

Step 4.1.15

Cancel the common factor of $4$ .

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Step 4.1.15.1

Factor $4$ out of $- 4$ .

$\frac{7203}{8} - \frac{7203}{8} + 4 (- 1) \frac{49}{4} + 24 (\frac{7}{2}) - 35$

Step 4.1.15.2

Cancel the common factor.

$\frac{7203}{8} - \frac{7203}{8} + 4 \cdot - 1 \frac{49}{4} + 24 (\frac{7}{2}) - 35$

Step 4.1.15.3

Rewrite the expression.

$\frac{7203}{8} - \frac{7203}{8} - 1 \cdot 49 + 24 (\frac{7}{2}) - 35$

Step 4.1.16

Multiply $- 1$ by $49$ .

$\frac{7203}{8} - \frac{7203}{8} - 49 + 24 (\frac{7}{2}) - 35$

Step 4.1.17

Cancel the common factor of $2$ .

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Step 4.1.17.1

Factor $2$ out of $24$ .

$\frac{7203}{8} - \frac{7203}{8} - 49 + 2 (12) \frac{7}{2} - 35$

Step 4.1.17.2

Cancel the common factor.

$\frac{7203}{8} - \frac{7203}{8} - 49 + 2 \cdot 12 \frac{7}{2} - 35$

Step 4.1.17.3

Rewrite the expression.

$\frac{7203}{8} - \frac{7203}{8} - 49 + 12 \cdot 7 - 35$

Step 4.1.18

Multiply $12$ by $7$ .

$\frac{7203}{8} - \frac{7203}{8} - 49 + 84 - 35$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$- 49 + 84 - 35 + \frac{7203 - 7203}{8}$

Step 4.2.2

Subtract $7203$ from $7203$ .

$- 49 + 84 - 35 + \frac{0}{8}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $- 49$ as a fraction with denominator $1$ .

$\frac{- 49}{1} + 84 - 35 + \frac{0}{8}$

Step 4.3.2

Multiply $\frac{- 49}{1}$ by $\frac{8}{8}$ .

$\frac{- 49}{1} \cdot \frac{8}{8} + 84 - 35 + \frac{0}{8}$

Step 4.3.3

Multiply $\frac{- 49}{1}$ by $\frac{8}{8}$ .

$\frac{- 49 \cdot 8}{8} + 84 - 35 + \frac{0}{8}$

Step 4.3.4

Write $84$ as a fraction with denominator $1$ .

$\frac{- 49 \cdot 8}{8} + \frac{84}{1} - 35 + \frac{0}{8}$

Step 4.3.5

Multiply $\frac{84}{1}$ by $\frac{8}{8}$ .

$\frac{- 49 \cdot 8}{8} + \frac{84}{1} \cdot \frac{8}{8} - 35 + \frac{0}{8}$

Step 4.3.6

Multiply $\frac{84}{1}$ by $\frac{8}{8}$ .

$\frac{- 49 \cdot 8}{8} + \frac{84 \cdot 8}{8} - 35 + \frac{0}{8}$

Step 4.3.7

Write $- 35$ as a fraction with denominator $1$ .

$\frac{- 49 \cdot 8}{8} + \frac{84 \cdot 8}{8} + \frac{- 35}{1} + \frac{0}{8}$

Step 4.3.8

Multiply $\frac{- 35}{1}$ by $\frac{8}{8}$ .

$\frac{- 49 \cdot 8}{8} + \frac{84 \cdot 8}{8} + \frac{- 35}{1} \cdot \frac{8}{8} + \frac{0}{8}$

Step 4.3.9

Multiply $\frac{- 35}{1}$ by $\frac{8}{8}$ .

$\frac{- 49 \cdot 8}{8} + \frac{84 \cdot 8}{8} + \frac{- 35 \cdot 8}{8} + \frac{0}{8}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 49 \cdot 8 + 84 \cdot 8 - 35 \cdot 8}{8}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $- 49$ by $8$ .

$\frac{- 392 + 84 \cdot 8 - 35 \cdot 8}{8}$

Step 4.5.2

Multiply $84$ by $8$ .

$\frac{- 392 + 672 - 35 \cdot 8}{8}$

Step 4.5.3

Multiply $- 35$ by $8$ .

$\frac{- 392 + 672 - 280}{8}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $- 392$ and $672$ .

$\frac{280 - 280}{8}$

Step 4.6.2

Subtract $280$ from $280$ .

$\frac{0}{8}$

Step 4.6.3

Divide $0$ by $8$ .

$0$

Step 5

Since $\frac{7}{2}$ is a known root, divide the polynomial by $x - \frac{7}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{6 x^{4} - 21 x^{3} - 4 x^{2} + 24 x - 35}{x - \frac{7}{2}}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$

Step 6.2

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$

	$6$

Step 6.3

Multiply the newest entry in the result $(6)$ by the divisor $(\frac{7}{2})$ and place the result of $(21)$ under the next term in the dividend $(- 21)$ .

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$
		$21$
	$6$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$
		$21$
	$6$	$0$

Step 6.5

Multiply the newest entry in the result $(0)$ by the divisor $(\frac{7}{2})$ and place the result of $(0)$ under the next term in the dividend $(- 4)$ .

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$
		$21$	$0$
	$6$	$0$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$
		$21$	$0$
	$6$	$0$	$- 4$

Step 6.7

Multiply the newest entry in the result $(- 4)$ by the divisor $(\frac{7}{2})$ and place the result of $(- 14)$ under the next term in the dividend $(24)$ .

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$
		$21$	$0$	$- 14$
	$6$	$0$	$- 4$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$
		$21$	$0$	$- 14$
	$6$	$0$	$- 4$	$10$

Step 6.9

Multiply the newest entry in the result $(10)$ by the divisor $(\frac{7}{2})$ and place the result of $(35)$ under the next term in the dividend $(- 35)$ .

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$
		$21$	$0$	$- 14$	$35$
	$6$	$0$	$- 4$	$10$

Step 6.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$\frac{7}{2}$	$6$	$- 21$	$- 4$	$24$	$- 35$
		$21$	$0$	$- 14$	$35$
	$6$	$0$	$- 4$	$10$	$0$

Step 6.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$6 x^{3} + 0 x^{2} + (- 4) x + 10$

Step 6.12

Simplify the quotient polynomial.

$6 x^{3} - 4 x + 10$

Step 7

Factor $2$ out of $6 x^{3} - 4 x + 10$ .

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Step 7.1

Factor $2$ out of $6 x^{3}$ .

$(x - \frac{7}{2}) (2 (3 x^{3}) - 4 x + 10)$

Step 7.2

Factor $2$ out of $- 4 x$ .

$(x - \frac{7}{2}) (2 (3 x^{3}) + 2 (- 2 x) + 10)$

Step 7.3

Factor $2$ out of $10$ .

$(x - \frac{7}{2}) (2 (3 x^{3}) + 2 (- 2 x) + 2 (5))$

Step 7.4

Factor $2$ out of $2 (3 x^{3}) + 2 (- 2 x)$ .

$(x - \frac{7}{2}) (2 (3 x^{3} - 2 x) + 2 (5))$

Step 7.5

Factor $2$ out of $2 (3 x^{3} - 2 x) + 2 (5)$ .

$(x - \frac{7}{2}) (2 (3 x^{3} - 2 x + 5))$

Step 8

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 1.37173942, \frac{7}{2}$

Step 9