Algebra Examples

$\frac{4}{x^{3}} - \frac{2 x - 1}{3 x}$

Step 1

To write $\frac{4}{x^{3}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4}{x^{3}} \cdot \frac{3}{3} - \frac{2 x - 1}{3 x}$

Step 2

To write $- \frac{2 x - 1}{3 x}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{4}{x^{3}} \cdot \frac{3}{3} - \frac{2 x - 1}{3 x} \cdot \frac{x^{2}}{x^{2}}$

Step 3

Write each expression with a common denominator of $3 x^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{4}{x^{3}}$ by $\frac{3}{3}$ .

$\frac{4 \cdot 3}{x^{3} \cdot 3} - \frac{2 x - 1}{3 x} \cdot \frac{x^{2}}{x^{2}}$

Step 3.2

Multiply $\frac{2 x - 1}{3 x}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{4 \cdot 3}{x^{3} \cdot 3} - \frac{(2 x - 1) x^{2}}{3 x \cdot x^{2}}$

Step 3.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.3.1

Move $x^{2}$ .

$\frac{4 \cdot 3}{x^{3} \cdot 3} - \frac{(2 x - 1) x^{2}}{3 (x^{2} x)}$

Step 3.3.2

Multiply $x^{2}$ by $x$ .

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Step 3.3.2.1

Raise $x$ to the power of $1$ .

$\frac{4 \cdot 3}{x^{3} \cdot 3} - \frac{(2 x - 1) x^{2}}{3 (x^{2} x^{1})}$

Step 3.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 \cdot 3}{x^{3} \cdot 3} - \frac{(2 x - 1) x^{2}}{3 x^{2 + 1}}$

Step 3.3.3

Add $2$ and $1$ .

$\frac{4 \cdot 3}{x^{3} \cdot 3} - \frac{(2 x - 1) x^{2}}{3 x^{3}}$

Step 3.4

Reorder the factors of $x^{3} \cdot 3$ .

$\frac{4 \cdot 3}{3 x^{3}} - \frac{(2 x - 1) x^{2}}{3 x^{3}}$

Step 4

Combine the numerators over the common denominator.

$\frac{4 \cdot 3 - (2 x - 1) x^{2}}{3 x^{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $4$ by $3$ .

$\frac{12 - (2 x - 1) x^{2}}{3 x^{3}}$

Step 5.2

Apply the distributive property.

$\frac{12 + (- (2 x) - - 1) x^{2}}{3 x^{3}}$

Step 5.3

Multiply $2$ by $- 1$ .

$\frac{12 + (- 2 x - - 1) x^{2}}{3 x^{3}}$

Step 5.4

Multiply $- 1$ by $- 1$ .

$\frac{12 + (- 2 x + 1) x^{2}}{3 x^{3}}$

Step 5.5

Apply the distributive property.

$\frac{12 - 2 x \cdot x^{2} + 1 x^{2}}{3 x^{3}}$

Step 5.6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.6.1

Move $x^{2}$ .

$\frac{12 - 2 (x^{2} x) + 1 x^{2}}{3 x^{3}}$

Step 5.6.2

Multiply $x^{2}$ by $x$ .

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Step 5.6.2.1

Raise $x$ to the power of $1$ .

$\frac{12 - 2 (x^{2} x^{1}) + 1 x^{2}}{3 x^{3}}$

Step 5.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{12 - 2 x^{2 + 1} + 1 x^{2}}{3 x^{3}}$

Step 5.6.3

Add $2$ and $1$ .

$\frac{12 - 2 x^{3} + 1 x^{2}}{3 x^{3}}$

Step 5.7

Multiply $x^{2}$ by $1$ .

$\frac{12 - 2 x^{3} + x^{2}}{3 x^{3}}$

Step 5.8

Reorder terms.

$\frac{- 2 x^{3} + x^{2} + 12}{3 x^{3}}$

Step 5.9

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

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Step 5.9.1

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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1, \pm 2$

Step 5.9.2

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

$\pm 1, \pm 0.5, \pm 12, \pm 6, \pm 2, \pm 3, \pm 1.5, \pm 4$

Step 5.9.3

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

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Step 5.9.3.1

Substitute $2$ into the polynomial.

$- 2 \cdot 2^{3} + 2^{2} + 12$

Step 5.9.3.2

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

$- 2 \cdot 8 + 2^{2} + 12$

Step 5.9.3.3

Multiply $- 2$ by $8$ .

$- 16 + 2^{2} + 12$

Step 5.9.3.4

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

$- 16 + 4 + 12$

Step 5.9.3.5

Add $- 16$ and $4$ .

$- 12 + 12$

Step 5.9.3.6

Add $- 12$ and $12$ .

$0$

Step 5.9.4

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.

$\frac{- 2 x^{3} + x^{2} + 12}{x - 2}$

Step 5.9.5

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

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Step 5.9.5.1

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

$x$

$2$

$2 x^{3}$

$x^{2}$

$0 x$

$12$

Step 5.9.5.2

Divide the highest order term in the dividend $- 2 x^{3}$ by the highest order term in divisor $x$ .

				-	$2 x^{2}$
	$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$

Step 5.9.5.3

Multiply the new quotient term by the divisor.

			-	$2 x^{2}$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			-	$2 x^{3}$	+	$4 x^{2}$

Step 5.9.5.4

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

			-	$2 x^{2}$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$

Step 5.9.5.5

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

			-	$2 x^{2}$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$

Step 5.9.5.6

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

			-	$2 x^{2}$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$

Step 5.9.5.7

Divide the highest order term in the dividend $- 3 x^{2}$ by the highest order term in divisor $x$ .

			-	$2 x^{2}$	-	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$

Step 5.9.5.8

Multiply the new quotient term by the divisor.

			-	$2 x^{2}$	-	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$
					-	$3 x^{2}$	+	$6 x$

Step 5.9.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{2} + 6 x$

			-	$2 x^{2}$	-	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$
					+	$3 x^{2}$	-	$6 x$

Step 5.9.5.10

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

			-	$2 x^{2}$	-	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$
					+	$3 x^{2}$	-	$6 x$
							-	$6 x$

Step 5.9.5.11

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

			-	$2 x^{2}$	-	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$
					+	$3 x^{2}$	-	$6 x$
							-	$6 x$	+	$12$

Step 5.9.5.12

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

			-	$2 x^{2}$	-	$3 x$	-	$6$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$
					+	$3 x^{2}$	-	$6 x$
							-	$6 x$	+	$12$

Step 5.9.5.13

Multiply the new quotient term by the divisor.

			-	$2 x^{2}$	-	$3 x$	-	$6$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$
					+	$3 x^{2}$	-	$6 x$
							-	$6 x$	+	$12$
							-	$6 x$	+	$12$

Step 5.9.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x + 12$

			-	$2 x^{2}$	-	$3 x$	-	$6$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$
					+	$3 x^{2}$	-	$6 x$
							-	$6 x$	+	$12$
							+	$6 x$	-	$12$

Step 5.9.5.15

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

			-	$2 x^{2}$	-	$3 x$	-	$6$
$x$	-	$2$	-	$2 x^{3}$	+	$x^{2}$	+	$0 x$	+	$12$
			+	$2 x^{3}$	-	$4 x^{2}$
					-	$3 x^{2}$	+	$0 x$
					+	$3 x^{2}$	-	$6 x$
							-	$6 x$	+	$12$
							+	$6 x$	-	$12$
										$0$

Step 5.9.5.16

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

$- 2 x^{2} - 3 x - 6$

Step 5.9.6

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

$\frac{(x - 2) (- 2 x^{2} - 3 x - 6)}{3 x^{3}}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 2 x^{2}$ .

$\frac{(x - 2) (- (2 x^{2}) - 3 x - 6)}{3 x^{3}}$

Step 6.2

Factor $- 1$ out of $- 3 x$ .

$\frac{(x - 2) (- (2 x^{2}) - (3 x) - 6)}{3 x^{3}}$

Step 6.3

Factor $- 1$ out of $- (2 x^{2}) - (3 x)$ .

$\frac{(x - 2) (- (2 x^{2} + 3 x) - 6)}{3 x^{3}}$

Step 6.4

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{(x - 2) (- (2 x^{2} + 3 x) - 1 (6))}{3 x^{3}}$

Step 6.5

Factor $- 1$ out of $- (2 x^{2} + 3 x) - 1 (6)$ .

$\frac{(x - 2) (- (2 x^{2} + 3 x + 6))}{3 x^{3}}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (2 x^{2} + 3 x + 6)$ as $- 1 (2 x^{2} + 3 x + 6)$ .

$\frac{(x - 2) (- 1 (2 x^{2} + 3 x + 6))}{3 x^{3}}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{(x - 2) (2 x^{2} + 3 x + 6)}{3 x^{3}}$