Precalculus Examples

$\frac{4 x^{3} + 5^{2} + 7 x - 1}{x^{2} x + 1}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor the fraction.

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Step 1.1.1

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

$\frac{4 x^{3} + 25 + 7 x - 1}{x^{2} x + 1}$

Step 1.1.2

Subtract $1$ from $25$ .

$\frac{4 x^{3} + 7 x + 24}{x^{2} x + 1}$

Step 1.1.3

Factor $4 x^{3} + 7 x + 24$ using the rational roots test.

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Step 1.1.3.1

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.

$p = \pm 1, \pm 24, \pm 2, \pm 12, \pm 3, \pm 8, \pm 4, \pm 6$

$q = \pm 1, \pm 4, \pm 2$

Step 1.1.3.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 24, \pm 6, \pm 12, \pm 2, \pm 3, \pm 0.75, \pm 1.5, \pm 8, \pm 4$

Step 1.1.3.3

Substitute $- 1.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1.5$ is a root of the polynomial.

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Step 1.1.3.3.1

Substitute $- 1.5$ into the polynomial.

$4 {(- 1.5)}^{3} + 7 \cdot - 1.5 + 24$

Step 1.1.3.3.2

Raise $- 1.5$ to the power of $3$ .

$4 \cdot - 3.375 + 7 \cdot - 1.5 + 24$

Step 1.1.3.3.3

Multiply $4$ by $- 3.375$ .

$- 13.5 + 7 \cdot - 1.5 + 24$

Step 1.1.3.3.4

Multiply $7$ by $- 1.5$ .

$- 13.5 - 10.5 + 24$

Step 1.1.3.3.5

Subtract $10.5$ from $- 13.5$ .

$- 24 + 24$

Step 1.1.3.3.6

Add $- 24$ and $24$ .

$0$

Step 1.1.3.4

Since $- 1.5$ is a known root, divide the polynomial by $2 x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} + 7 x + 24}{2 x + 3}$

Step 1.1.3.5

Divide $4 x^{3} + 7 x + 24$ by $2 x + 3$ .

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Step 1.1.3.5.1

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

$2 x$

$3$

$4 x^{3}$

$0 x^{2}$

$7 x$

$24$

Step 1.1.3.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$

Step 1.1.3.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			+	$4 x^{3}$	+	$6 x^{2}$

Step 1.1.3.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 6 x^{2}$

				$2 x^{2}$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$

Step 1.1.3.5.5

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

				$2 x^{2}$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$

Step 1.1.3.5.6

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

				$2 x^{2}$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$

Step 1.1.3.5.7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$

Step 1.1.3.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					-	$6 x^{2}$	-	$9 x$

Step 1.1.3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 9 x$

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$9 x$

Step 1.1.3.5.10

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

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$9 x$
							+	$16 x$

Step 1.1.3.5.11

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

				$2 x^{2}$	-	$3 x$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$9 x$
							+	$16 x$	+	$24$

Step 1.1.3.5.12

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$3 x$	+	$8$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$9 x$
							+	$16 x$	+	$24$

Step 1.1.3.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$	+	$8$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$9 x$
							+	$16 x$	+	$24$
							+	$16 x$	+	$24$

Step 1.1.3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x + 24$

				$2 x^{2}$	-	$3 x$	+	$8$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$9 x$
							+	$16 x$	+	$24$
							-	$16 x$	-	$24$

Step 1.1.3.5.15

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

				$2 x^{2}$	-	$3 x$	+	$8$
$2 x$	+	$3$		$4 x^{3}$	+	$0 x^{2}$	+	$7 x$	+	$24$
			-	$4 x^{3}$	-	$6 x^{2}$
					-	$6 x^{2}$	+	$7 x$
					+	$6 x^{2}$	+	$9 x$
							+	$16 x$	+	$24$
							-	$16 x$	-	$24$
										$0$

Step 1.1.3.5.16

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

$2 x^{2} - 3 x + 8$

Step 1.1.3.6

Write $4 x^{3} + 7 x + 24$ as a set of factors.

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8)}{x^{2} x + 1}$

Step 1.1.4

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

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Step 1.1.4.1

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

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Step 1.1.4.1.1

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

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8)}{x^{2} x^{1} + 1}$

Step 1.1.4.1.2

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

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8)}{x^{2 + 1} + 1}$

Step 1.1.4.2

Add $2$ and $1$ .

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8)}{x^{3} + 1}$

Step 1.1.5

Rewrite $1$ as $1^{3}$ .

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8)}{x^{3} + 1^{3}}$

Step 1.1.6

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8)}{(x + 1) (x^{2} - x \cdot 1 + 1^{2})}$

Step 1.1.7

Simplify.

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Step 1.1.7.1

Multiply $- 1$ by $1$ .

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8)}{(x + 1) (x^{2} - x + 1^{2})}$

Step 1.1.7.2

One to any power is one.

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8)}{(x + 1) (x^{2} - x + 1)}$

Step 1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{x + 1}$

Step 1.3

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor is 2nd order, $2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

$\frac{A}{x + 1} + \frac{B x + C}{x^{2} - x + 1}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x + 1) (x^{2} - x + 1)$ .

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8) (x + 1) (x^{2} - x + 1)}{(x + 1) (x^{2} - x + 1)} = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.5

Reduce the expression by cancelling the common factors.

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Step 1.5.1

Cancel the common factor of $x + 1$ .

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Step 1.5.1.1

Cancel the common factor.

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8) (x + 1) (x^{2} - x + 1)}{(x + 1) (x^{2} - x + 1)} = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.5.1.2

Rewrite the expression.

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8) (x^{2} - x + 1)}{x^{2} - x + 1} = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.5.2

Cancel the common factor of $x^{2} - x + 1$ .

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Step 1.5.2.1

Cancel the common factor.

$\frac{(2 x + 3) (2 x^{2} - 3 x + 8) (x^{2} - x + 1)}{x^{2} - x + 1} = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.5.2.2

Divide $(2 x + 3) (2 x^{2} - 3 x + 8)$ by $1$ .

$(2 x + 3) (2 x^{2} - 3 x + 8) = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.6

Expand $(2 x + 3) (2 x^{2} - 3 x + 8)$ by multiplying each term in the first expression by each term in the second expression.

$2 x (2 x^{2}) + 2 x (- 3 x) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7

Simplify terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot (2 x \cdot x^{2}) + 2 x (- 3 x) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.2

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

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Step 1.7.1.2.1

Move $x^{2}$ .

$2 \cdot (2 (x^{2} x)) + 2 x (- 3 x) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.2.2

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

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Step 1.7.1.2.2.1

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

$2 \cdot (2 (x^{2} x)) + 2 x (- 3 x) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.2.2.2

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

$2 \cdot (2 x^{2 + 1}) + 2 x (- 3 x) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.2.3

Add $2$ and $1$ .

$2 \cdot (2 x^{3}) + 2 x (- 3 x) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.3

Multiply $2$ by $2$ .

$4 x^{3} + 2 x (- 3 x) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.4

Rewrite using the commutative property of multiplication.

$4 x^{3} + 2 \cdot (- 3 x \cdot x) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 1.7.1.5.1

Move $x$ .

$4 x^{3} + 2 \cdot (- 3 (x \cdot x)) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.5.2

Multiply $x$ by $x$ .

$4 x^{3} + 2 \cdot (- 3 x^{2}) + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.6

Multiply $2$ by $- 3$ .

$4 x^{3} - 6 x^{2} + 2 x \cdot 8 + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.7

Multiply $8$ by $2$ .

$4 x^{3} - 6 x^{2} + 16 x + 3 (2 x^{2}) + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.8

Multiply $2$ by $3$ .

$4 x^{3} - 6 x^{2} + 16 x + 6 x^{2} + 3 (- 3 x) + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.9

Multiply $- 3$ by $3$ .

$4 x^{3} - 6 x^{2} + 16 x + 6 x^{2} - 9 x + 3 \cdot 8 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.1.10

Multiply $3$ by $8$ .

$4 x^{3} - 6 x^{2} + 16 x + 6 x^{2} - 9 x + 24 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.2

Simplify by adding terms.

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Step 1.7.2.1

Combine the opposite terms in $4 x^{3} - 6 x^{2} + 16 x + 6 x^{2} - 9 x + 24$ .

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Step 1.7.2.1.1

Add $- 6 x^{2}$ and $6 x^{2}$ .

$4 x^{3} + 16 x + 0 - 9 x + 24 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.2.1.2

Add $4 x^{3} + 16 x$ and $0$ .

$4 x^{3} + 16 x - 9 x + 24 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.7.2.2

Subtract $9 x$ from $16 x$ .

$4 x^{3} + 7 x + 24 = \frac{(A) (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.8

Simplify each term.

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Step 1.8.1

Cancel the common factor of $x + 1$ .

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Step 1.8.1.1

Cancel the common factor.

$4 x^{3} + 7 x + 24 = \frac{A (x + 1) (x^{2} - x + 1)}{x + 1} + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.8.1.2

Divide $(A) (x^{2} - x + 1)$ by $1$ .

$4 x^{3} + 7 x + 24 = (A) (x^{2} - x + 1) + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.8.2

Apply the distributive property.

$4 x^{3} + 7 x + 24 = A x^{2} + A (- x) + A \cdot 1 + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.8.3

Simplify.

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Step 1.8.3.1

Rewrite using the commutative property of multiplication.

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A \cdot 1 + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.8.3.2

Multiply $A$ by $1$ .

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.8.4

Cancel the common factor of $x^{2} - x + 1$ .

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Step 1.8.4.1

Cancel the common factor.

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + \frac{(B x + C) (x + 1) (x^{2} - x + 1)}{x^{2} - x + 1}$

Step 1.8.4.2

Divide $(B x + C) (x + 1)$ by $1$ .

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + (B x + C) (x + 1)$

Step 1.8.5

Expand $(B x + C) (x + 1)$ using the FOIL Method.

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Step 1.8.5.1

Apply the distributive property.

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + B x (x + 1) + C (x + 1)$

Step 1.8.5.2

Apply the distributive property.

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + B x \cdot x + B x \cdot 1 + C (x + 1)$

Step 1.8.5.3

Apply the distributive property.

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + B x \cdot x + B x \cdot 1 + C x + C \cdot 1$

Step 1.8.6

Simplify each term.

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Step 1.8.6.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.8.6.1.1

Move $x$ .

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + B (x \cdot x) + B x \cdot 1 + C x + C \cdot 1$

Step 1.8.6.1.2

Multiply $x$ by $x$ .

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + B x^{2} + B x \cdot 1 + C x + C \cdot 1$

Step 1.8.6.2

Multiply $B$ by $1$ .

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + B x^{2} + B x + C x + C \cdot 1$

Step 1.8.6.3

Multiply $C$ by $1$ .

$4 x^{3} + 7 x + 24 = A x^{2} - A x + A + B x^{2} + B x + C x + C$

Step 1.9

Simplify the expression.

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Step 1.9.1

Move $A$ .

$4 x^{3} + 7 x + 24 = A x^{2} - x A + A + B x^{2} + B x + C x + C$

Step 1.9.2

Reorder $B$ and $x^{2}$ .

$4 x^{3} + 7 x + 24 = A x^{2} - x A + A + B x^{2} + B x + C x + C$

Step 1.9.3

Reorder $B$ and $x$ .

$4 x^{3} + 7 x + 24 = A x^{2} - x A + A + B x^{2} + B x + C x + C$

Step 1.9.4

Move $A$ .

$4 x^{3} + 7 x + 24 = A x^{2} - x A + B x^{2} + B x + C x + A + C$

Step 1.9.5

Move $- x A$ .

$4 x^{3} + 7 x + 24 = A x^{2} + B x^{2} - x A + B x + C x + A + C$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A + B$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$7 = - 1 A + B + C$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$24 = A + C$

Step 2.4

Set up the system of equations to find the coefficients of the partial fractions.

$0 = A + B$

$7 = - 1 A + B + C$

$24 = A + C$

$0 = A + B$

$7 = - 1 A + B + C$

$24 = A + C$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $0 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 0$ .

$A + B = 0$

$7 = - 1 A + B + C$

$24 = A + C$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = - B$

$7 = - 1 A + B + C$

$24 = A + C$

$A = - B$

$7 = - 1 A + B + C$

$24 = A + C$

Step 3.2

Replace all occurrences of $A$ with $- B$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $7 = - 1 A + B + C$ with $- B$ .

$7 = - 1 (- B) + B + C$

$A = - B$

$24 = A + C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 1 (- B) + B + C$ .

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Step 3.2.2.1.1

Multiply $- 1 (- B)$ .

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Step 3.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$7 = 1 B + B + C$

$A = - B$

$24 = A + C$

Step 3.2.2.1.1.2

Multiply $B$ by $1$ .

$7 = B + B + C$

$A = - B$

$24 = A + C$

$7 = B + B + C$

$A = - B$

$24 = A + C$

Step 3.2.2.1.2

Add $B$ and $B$ .

$7 = 2 B + C$

$A = - B$

$24 = A + C$

$7 = 2 B + C$

$A = - B$

$24 = A + C$

$7 = 2 B + C$

$A = - B$

$24 = A + C$

Step 3.2.3

Replace all occurrences of $A$ in $24 = A + C$ with $- B$ .

$24 = (- B) + C$

$7 = 2 B + C$

$A = - B$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Remove parentheses.

$24 = - B + C$

$7 = 2 B + C$

$A = - B$

$24 = - B + C$

$7 = 2 B + C$

$A = - B$

$24 = - B + C$

$7 = 2 B + C$

$A = - B$

Step 3.3

Solve for $C$ in $24 = - B + C$ .

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Step 3.3.1

Rewrite the equation as $- B + C = 24$ .

$- B + C = 24$

$7 = 2 B + C$

$A = - B$

Step 3.3.2

Add $B$ to both sides of the equation.

$C = 24 + B$

$7 = 2 B + C$

$A = - B$

$C = 24 + B$

$7 = 2 B + C$

$A = - B$

Step 3.4

Replace all occurrences of $C$ with $24 + B$ in each equation.

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Step 3.4.1

Replace all occurrences of $C$ in $7 = 2 B + C$ with $24 + B$ .

$7 = 2 B + 24 + B$

$C = 24 + B$

$A = - B$

Step 3.4.2

Simplify $7 = 2 B + 24 + B$ .

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Step 3.4.2.1

Simplify the left side.

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Step 3.4.2.1.1

Remove parentheses.

$7 = 2 B + 24 + B$

$C = 24 + B$

$A = - B$

$7 = 2 B + 24 + B$

$C = 24 + B$

$A = - B$

Step 3.4.2.2

Simplify the right side.

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Step 3.4.2.2.1

Add $2 B$ and $B$ .

$7 = 3 B + 24$

$C = 24 + B$

$A = - B$

$7 = 3 B + 24$

$C = 24 + B$

$A = - B$

$7 = 3 B + 24$

$C = 24 + B$

$A = - B$

$7 = 3 B + 24$

$C = 24 + B$

$A = - B$

Step 3.5

Solve for $B$ in $7 = 3 B + 24$ .

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Step 3.5.1

Rewrite the equation as $3 B + 24 = 7$ .

$3 B + 24 = 7$

$C = 24 + B$

$A = - B$

Step 3.5.2

Move all terms not containing $B$ to the right side of the equation.

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Step 3.5.2.1

Subtract $24$ from both sides of the equation.

$3 B = 7 - 24$

$C = 24 + B$

$A = - B$

Step 3.5.2.2

Subtract $24$ from $7$ .

$3 B = - 17$

$C = 24 + B$

$A = - B$

$3 B = - 17$

$C = 24 + B$

$A = - B$

Step 3.5.3

Divide each term in $3 B = - 17$ by $3$ and simplify.

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Step 3.5.3.1

Divide each term in $3 B = - 17$ by $3$ .

$\frac{3 B}{3} = \frac{- 17}{3}$

$C = 24 + B$

$A = - B$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $3$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{3 B}{3} = \frac{- 17}{3}$

$C = 24 + B$

$A = - B$

Step 3.5.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 17}{3}$

$C = 24 + B$

$A = - B$

$B = \frac{- 17}{3}$

$C = 24 + B$

$A = - B$

$B = \frac{- 17}{3}$

$C = 24 + B$

$A = - B$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Move the negative in front of the fraction.

$B = - \frac{17}{3}$

$C = 24 + B$

$A = - B$

$B = - \frac{17}{3}$

$C = 24 + B$

$A = - B$

$B = - \frac{17}{3}$

$C = 24 + B$

$A = - B$

$B = - \frac{17}{3}$

$C = 24 + B$

$A = - B$

Step 3.6

Replace all occurrences of $B$ with $- \frac{17}{3}$ in each equation.

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Step 3.6.1

Replace all occurrences of $B$ in $C = 24 + B$ with $- \frac{17}{3}$ .

$C = 24 - \frac{17}{3}$

$B = - \frac{17}{3}$

$A = - B$

Step 3.6.2

Simplify $C = 24 - \frac{17}{3}$ .

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Step 3.6.2.1

Simplify the left side.

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Step 3.6.2.1.1

Remove parentheses.

$C = 24 - \frac{17}{3}$

$B = - \frac{17}{3}$

$A = - B$

$C = 24 - \frac{17}{3}$

$B = - \frac{17}{3}$

$A = - B$

Step 3.6.2.2

Simplify the right side.

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Step 3.6.2.2.1

Simplify $24 - \frac{17}{3}$ .

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Step 3.6.2.2.1.1

To write $24$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$C = 24 \cdot \frac{3}{3} - \frac{17}{3}$

$B = - \frac{17}{3}$

$A = - B$

Step 3.6.2.2.1.2

Combine $24$ and $\frac{3}{3}$ .

$C = \frac{24 \cdot 3}{3} - \frac{17}{3}$

$B = - \frac{17}{3}$

$A = - B$

Step 3.6.2.2.1.3

Combine the numerators over the common denominator.

$C = \frac{24 \cdot 3 - 17}{3}$

$B = - \frac{17}{3}$

$A = - B$

Step 3.6.2.2.1.4

Simplify the numerator.

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Step 3.6.2.2.1.4.1

Multiply $24$ by $3$ .

$C = \frac{72 - 17}{3}$

$B = - \frac{17}{3}$

$A = - B$

Step 3.6.2.2.1.4.2

Subtract $17$ from $72$ .

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

$A = - B$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

$A = - B$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

$A = - B$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

$A = - B$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

$A = - B$

Step 3.6.3

Replace all occurrences of $B$ in $A = - B$ with $- \frac{17}{3}$ .

$A = - (- \frac{17}{3})$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Multiply $- (- \frac{17}{3})$ .

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Step 3.6.4.1.1

Multiply $- 1$ by $- 1$ .

$A = 1 (\frac{17}{3})$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

Step 3.6.4.1.2

Multiply $\frac{17}{3}$ by $1$ .

$A = \frac{17}{3}$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

$A = \frac{17}{3}$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

$A = \frac{17}{3}$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

$A = \frac{17}{3}$

$C = \frac{55}{3}$

$B = - \frac{17}{3}$

Step 3.7

List all of the solutions.

$A = \frac{17}{3}, C = \frac{55}{3}, B = - \frac{17}{3}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x + 1} + \frac{B x + C}{x^{2} - x + 1}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{\frac{17}{3}}{x + 1} + \frac{- \frac{17}{3 x} + \frac{55}{3}}{x^{2} - x + 1}$

Step 5

Simplify each term.

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Step 5.1

Combine $x$ and $\frac{17}{3}$ .

$\frac{\frac{17}{3}}{x + 1} + \frac{- \frac{x \cdot 17}{3} + \frac{55}{3}}{x^{2} - x + 1}$

Step 5.2

Move $17$ to the left of $x$ .

$\frac{\frac{17}{3}}{x + 1} + \frac{- \frac{17 x}{3} + \frac{55}{3}}{x^{2} - x + 1}$