Precalculus Examples

$\frac{- 10 x^{2} + 27 x - 14}{{(x - 1)}^{3} (x + 2)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor by grouping.

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Step 1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 10 \cdot - 14 = 140$ and whose sum is $b = 27$ .

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Step 1.1.1.1

Factor $27$ out of $27 x$ .

$\frac{- 10 x^{2} + 27 (x) - 14}{{(x - 1)}^{3} (x + 2)}$

Step 1.1.1.2

Rewrite $27$ as $7$ plus $20$

$\frac{- 10 x^{2} + (7 + 20) x - 14}{{(x - 1)}^{3} (x + 2)}$

Step 1.1.1.3

Apply the distributive property.

$\frac{- 10 x^{2} + 7 x + 20 x - 14}{{(x - 1)}^{3} (x + 2)}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{(- 10 x^{2} + 7 x) + 20 x - 14}{{(x - 1)}^{3} (x + 2)}$

Step 1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- 10 x + 7) - 2 (- 10 x + 7)}{{(x - 1)}^{3} (x + 2)}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $- 10 x + 7$ .

$\frac{(- 10 x + 7) (x - 2)}{{(x - 1)}^{3} (x + 2)}$

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 $A$ .

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

Step 1.3

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

Step 1.4

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

Step 1.5

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

Step 1.6

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(x - 1)}^{3} (x + 2)$ .

$\frac{(- 10 x + 7) (x - 2) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3} (x + 2)} = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.7

Reduce the expression by cancelling the common factors.

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Step 1.7.1

Cancel the common factor of ${(x - 1)}^{3}$ .

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Step 1.7.1.1

Cancel the common factor.

Step 1.7.1.2

Rewrite the expression.

$\frac{(- 10 x + 7) (x - 2) (x + 2)}{x + 2} = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.7.2

Cancel the common factor of $x + 2$ .

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Step 1.7.2.1

Cancel the common factor.

Step 1.7.2.2

Divide $(- 10 x + 7) (x - 2)$ by $1$ .

$(- 10 x + 7) (x - 2) = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.8

Expand $(- 10 x + 7) (x - 2)$ using the FOIL Method.

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Step 1.8.1

Apply the distributive property.

$- 10 x (x - 2) + 7 (x - 2) = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.8.2

Apply the distributive property.

$- 10 x \cdot x - 10 x \cdot - 2 + 7 (x - 2) = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.8.3

Apply the distributive property.

$- 10 x \cdot x - 10 x \cdot - 2 + 7 x + 7 \cdot - 2 = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.9

Simplify and combine like terms.

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

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

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Step 1.9.1.1.1

Move $x$ .

$- 10 (x \cdot x) - 10 x \cdot - 2 + 7 x + 7 \cdot - 2 = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.9.1.1.2

Multiply $x$ by $x$ .

$- 10 x^{2} - 10 x \cdot - 2 + 7 x + 7 \cdot - 2 = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.9.1.2

Multiply $- 2$ by $- 10$ .

$- 10 x^{2} + 20 x + 7 x + 7 \cdot - 2 = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.9.1.3

Multiply $7$ by $- 2$ .

$- 10 x^{2} + 20 x + 7 x - 14 = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.9.2

Add $20 x$ and $7 x$ .

$- 10 x^{2} + 27 x - 14 = \frac{(A) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10

Simplify each term.

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Step 1.10.1

Cancel the common factor of ${(x - 1)}^{3}$ .

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Step 1.10.1.1

Cancel the common factor.

$- 10 x^{2} + 27 x - 14 = \frac{A {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{3}} + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.1.2

Divide $(A) (x + 2)$ by $1$ .

$- 10 x^{2} + 27 x - 14 = (A) (x + 2) + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.2

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + A \cdot 2 + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.3

Move $2$ to the left of $A$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 \cdot A + \frac{(B) {(x - 1)}^{3} (x + 2)}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.4

Cancel the common factor of ${(x - 1)}^{3}$ and ${(x - 1)}^{2}$ .

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Step 1.10.4.1

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

$- 10 x^{2} + 27 x - 14 = A x + 2 A + \frac{{(x - 1)}^{2} (B (x - 1) (x + 2))}{{(x - 1)}^{2}} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.4.2

Cancel the common factors.

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Step 1.10.4.2.1

Multiply by $1$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + \frac{{(x - 1)}^{2} (B (x - 1) (x + 2))}{{(x - 1)}^{2} \cdot 1} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.4.2.2

Cancel the common factor.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + \frac{{(x - 1)}^{2} (B (x - 1) (x + 2))}{{(x - 1)}^{2} \cdot 1} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.4.2.3

Rewrite the expression.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + \frac{B (x - 1) (x + 2)}{1} + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.4.2.4

Divide $B (x - 1) (x + 2)$ by $1$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B (x - 1) (x + 2) + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.5

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + (B x + B \cdot - 1) (x + 2) + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.6

Move $- 1$ to the left of $B$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + (B x - 1 \cdot B) (x + 2) + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.7

Rewrite $- 1 B$ as $- B$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + (B x - B) (x + 2) + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.8

Expand $(B x - B) (x + 2)$ using the FOIL Method.

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Step 1.10.8.1

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x (x + 2) - B (x + 2) + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.8.2

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x \cdot x + B x \cdot 2 - B (x + 2) + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.8.3

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x \cdot x + B x \cdot 2 - B x - B \cdot 2 + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.9

Simplify and combine like terms.

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Step 1.10.9.1

Simplify each term.

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Step 1.10.9.1.1

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

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Step 1.10.9.1.1.1

Move $x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B (x \cdot x) + B x \cdot 2 - B x - B \cdot 2 + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.9.1.1.2

Multiply $x$ by $x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x \cdot 2 - B x - B \cdot 2 + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.9.1.2

Move $2$ to the left of $B x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + 2 \cdot (B x) - B x - B \cdot 2 + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.9.1.3

Multiply $2$ by $- 1$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + 2 B x - B x - 2 B + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.9.2

Subtract $B x$ from $2 B x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + 1 B x - 2 B + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.10

Multiply $B$ by $1$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + \frac{(C) {(x - 1)}^{3} (x + 2)}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.11

Cancel the common factor of ${(x - 1)}^{3}$ and $x - 1$ .

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Step 1.10.11.1

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

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + \frac{(x - 1) (C {(x - 1)}^{2} (x + 2))}{x - 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.11.2

Cancel the common factors.

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Step 1.10.11.2.1

Multiply by $1$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + \frac{(x - 1) (C {(x - 1)}^{2} (x + 2))}{(x - 1) \cdot 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.11.2.2

Cancel the common factor.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + \frac{(x - 1) (C {(x - 1)}^{2} (x + 2))}{(x - 1) \cdot 1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.11.2.3

Rewrite the expression.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + \frac{C {(x - 1)}^{2} (x + 2)}{1} + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.11.2.4

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

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C {(x - 1)}^{2} (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.12

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

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C ((x - 1) (x - 1)) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.13

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.10.13.1

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x (x - 1) - 1 (x - 1)) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.13.2

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x \cdot x + x \cdot - 1 - 1 (x - 1)) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.13.3

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.14

Simplify and combine like terms.

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Step 1.10.14.1

Simplify each term.

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Step 1.10.14.1.1

Multiply $x$ by $x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.14.1.2

Move $- 1$ to the left of $x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.14.1.3

Rewrite $- 1 x$ as $- x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x^{2} - x - 1 x - 1 \cdot - 1) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.14.1.4

Rewrite $- 1 x$ as $- x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x^{2} - x - x - 1 \cdot - 1) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.14.1.5

Multiply $- 1$ by $- 1$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x^{2} - x - x + 1) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.14.2

Subtract $x$ from $- x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x^{2} - 2 x + 1) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.15

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + (C x^{2} + C (- 2 x) + C \cdot 1) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.16

Simplify.

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Step 1.10.16.1

Rewrite using the commutative property of multiplication.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + (C x^{2} - 2 C x + C \cdot 1) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.16.2

Multiply $C$ by $1$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + (C x^{2} - 2 C x + C) (x + 2) + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.17

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

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{2} x + C x^{2} \cdot 2 - 2 C x \cdot x - 2 C x \cdot 2 + C x + C \cdot 2 + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.18

Simplify each term.

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Step 1.10.18.1

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

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Step 1.10.18.1.1

Move $x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x \cdot x^{2}) + C x^{2} \cdot 2 - 2 C x \cdot x - 2 C x \cdot 2 + C x + C \cdot 2 + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.18.1.2

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

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Step 1.10.18.1.2.1

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

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C (x \cdot x^{2}) + C x^{2} \cdot 2 - 2 C x \cdot x - 2 C x \cdot 2 + C x + C \cdot 2 + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.18.1.2.2

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

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{1 + 2} + C x^{2} \cdot 2 - 2 C x \cdot x - 2 C x \cdot 2 + C x + C \cdot 2 + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.18.1.3

Add $1$ and $2$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} + C x^{2} \cdot 2 - 2 C x \cdot x - 2 C x \cdot 2 + C x + C \cdot 2 + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.18.2

Move $2$ to the left of $C x^{2}$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} + 2 \cdot (C x^{2}) - 2 C x \cdot x - 2 C x \cdot 2 + C x + C \cdot 2 + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.18.3

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

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Step 1.10.18.3.1

Move $x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} + 2 C x^{2} - 2 C (x \cdot x) - 2 C x \cdot 2 + C x + C \cdot 2 + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.18.3.2

Multiply $x$ by $x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} + 2 C x^{2} - 2 C x^{2} - 2 C x \cdot 2 + C x + C \cdot 2 + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.18.4

Multiply $2$ by $- 2$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} + 2 C x^{2} - 2 C x^{2} - 4 C x + C x + C \cdot 2 + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.18.5

Move $2$ to the left of $C$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} + 2 C x^{2} - 2 C x^{2} - 4 C x + C x + 2 C + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.19

Combine the opposite terms in $C x^{3} + 2 C x^{2} - 2 C x^{2} - 4 C x + C x + 2 C$ .

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Step 1.10.19.1

Subtract $2 C x^{2}$ from $2 C x^{2}$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} + 0 - 4 C x + C x + 2 C + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.19.2

Add $C x^{3}$ and $0$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 4 C x + C x + 2 C + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.20

Add $- 4 C x$ and $C x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.21

Cancel the common factor of $x + 2$ .

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Step 1.10.21.1

Cancel the common factor.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + \frac{(D) {(x - 1)}^{3} (x + 2)}{x + 2}$

Step 1.10.21.2

Divide $(D) {(x - 1)}^{3}$ by $1$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + (D) {(x - 1)}^{3}$

Step 1.10.22

Use the Binomial Theorem.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D (x^{3} + 3 x^{2} \cdot - 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.10.23

Simplify each term.

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Step 1.10.23.1

Multiply $- 1$ by $3$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D (x^{3} - 3 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.10.23.2

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

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D (x^{3} - 3 x^{2} + 3 x \cdot 1 + {(- 1)}^{3})$

Step 1.10.23.3

Multiply $3$ by $1$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D (x^{3} - 3 x^{2} + 3 x + {(- 1)}^{3})$

Step 1.10.23.4

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

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D (x^{3} - 3 x^{2} + 3 x - 1)$

Step 1.10.24

Apply the distributive property.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D x^{3} + D (- 3 x^{2}) + D (3 x) + D \cdot - 1$

Step 1.10.25

Simplify.

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Step 1.10.25.1

Rewrite using the commutative property of multiplication.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D x^{3} - 3 D x^{2} + D (3 x) + D \cdot - 1$

Step 1.10.25.2

Rewrite using the commutative property of multiplication.

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D x^{3} - 3 D x^{2} + 3 D x + D \cdot - 1$

Step 1.10.25.3

Move $- 1$ to the left of $D$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D x^{3} - 3 D x^{2} + 3 D x - 1 \cdot D$

Step 1.10.26

Rewrite $- 1 D$ as $- D$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D x^{3} - 3 D x^{2} + 3 D x - D$

Step 1.11

Simplify the expression.

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Step 1.11.1

Reorder $B$ and $x$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + 2 C + D x^{3} - 3 D x^{2} + 3 D x - D$

Step 1.11.2

Move $2 C$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x - 2 B + C x^{3} - 3 C x + D x^{3} - 3 D x^{2} + 3 D x + 2 C - D$

Step 1.11.3

Move $- 2 B$ .

$- 10 x^{2} + 27 x - 14 = A x + 2 A + B x^{2} + B x + C x^{3} - 3 C x + D x^{3} - 3 D x^{2} + 3 D x - 2 B + 2 C - D$

Step 1.11.4

Move $2 A$ .

$- 10 x^{2} + 27 x - 14 = A x + B x^{2} + B x + C x^{3} - 3 C x + D x^{3} - 3 D x^{2} + 3 D x + 2 A - 2 B + 2 C - D$

Step 1.11.5

Move $- 3 C x$ .

$- 10 x^{2} + 27 x - 14 = A x + B x^{2} + B x + C x^{3} + D x^{3} - 3 D x^{2} - 3 C x + 3 D x + 2 A - 2 B + 2 C - D$

Step 1.11.6

Move $B x$ .

$- 10 x^{2} + 27 x - 14 = A x + B x^{2} + C x^{3} + D x^{3} - 3 D x^{2} + B x - 3 C x + 3 D x + 2 A - 2 B + 2 C - D$

Step 1.11.7

Move $A x$ .

$- 10 x^{2} + 27 x - 14 = B x^{2} + C x^{3} + D x^{3} - 3 D x^{2} + A x + B x - 3 C x + 3 D x + 2 A - 2 B + 2 C - D$

Step 1.11.8

Move $B x^{2}$ .

$- 10 x^{2} + 27 x - 14 = C x^{3} + D x^{3} + B x^{2} - 3 D x^{2} + A x + B x - 3 C x + 3 D x + 2 A - 2 B + 2 C - D$

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 $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = C + D$

Step 2.2

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.

$- 10 = B - 3 D$

Step 2.3

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.

$27 = A + B - 3 C + 3 D$

Step 2.4

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.

$- 14 = 2 A - 2 B + 2 C - 1 D$

Step 2.5

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

$0 = C + D$

$- 10 = B - 3 D$

$27 = A + B - 3 C + 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

$0 = C + D$

$- 10 = B - 3 D$

$27 = A + B - 3 C + 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $C$ in $0 = C + D$ .

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Step 3.1.1

Rewrite the equation as $C + D = 0$ .

$C + D = 0$

$- 10 = B - 3 D$

$27 = A + B - 3 C + 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

Step 3.1.2

Subtract $D$ from both sides of the equation.

$C = - D$

$- 10 = B - 3 D$

$27 = A + B - 3 C + 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

$C = - D$

$- 10 = B - 3 D$

$27 = A + B - 3 C + 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

Step 3.2

Replace all occurrences of $C$ with $- D$ in each equation.

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Step 3.2.1

Replace all occurrences of $C$ in $27 = A + B - 3 C + 3 D$ with $- D$ .

$27 = A + B - 3 (- D) + 3 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $A + B - 3 (- D) + 3 D$ .

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Step 3.2.2.1.1

Multiply $- 1$ by $- 3$ .

$27 = A + B + 3 D + 3 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

Step 3.2.2.1.2

Add $3 D$ and $3 D$ .

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 2 A - 2 B + 2 C - 1 D$

Step 3.2.3

Replace all occurrences of $C$ in $- 14 = 2 A - 2 B + 2 C - 1 D$ with $- D$ .

$- 14 = 2 A - 2 B + 2 (- D) - 1 D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify $2 A - 2 B + 2 (- D) - 1 D$ .

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Step 3.2.4.1.1

Simplify each term.

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Step 3.2.4.1.1.1

Multiply $- 1$ by $2$ .

$- 14 = 2 A - 2 B - 2 D - 1 D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.2.4.1.1.2

Rewrite $- 1 D$ as $- D$ .

$- 14 = 2 A - 2 B - 2 D - D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 2 A - 2 B - 2 D - D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.2.4.1.2

Subtract $D$ from $- 2 D$ .

$- 14 = 2 A - 2 B - 3 D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 2 A - 2 B - 3 D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 2 A - 2 B - 3 D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 2 A - 2 B - 3 D$

$27 = A + B + 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.3

Solve for $A$ in $27 = A + B + 6 D$ .

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Step 3.3.1

Rewrite the equation as $A + B + 6 D = 27$ .

$A + B + 6 D = 27$

$- 14 = 2 A - 2 B - 3 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $B$ from both sides of the equation.

$A + 6 D = 27 - B$

$- 14 = 2 A - 2 B - 3 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.3.2.2

Subtract $6 D$ from both sides of the equation.

$A = 27 - B - 6 D$

$- 14 = 2 A - 2 B - 3 D$

$C = - D$

$- 10 = B - 3 D$

$A = 27 - B - 6 D$

$- 14 = 2 A - 2 B - 3 D$

$C = - D$

$- 10 = B - 3 D$

$A = 27 - B - 6 D$

$- 14 = 2 A - 2 B - 3 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.4

Replace all occurrences of $A$ with $27 - B - 6 D$ in each equation.

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Step 3.4.1

Replace all occurrences of $A$ in $- 14 = 2 A - 2 B - 3 D$ with $27 - B - 6 D$ .

$- 14 = 2 (27 - B - 6 D) - 2 B - 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $2 (27 - B - 6 D) - 2 B - 3 D$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$- 14 = 2 \cdot 27 + 2 (- B) + 2 (- 6 D) - 2 B - 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.4.2.1.1.2

Simplify.

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Step 3.4.2.1.1.2.1

Multiply $2$ by $27$ .

$- 14 = 54 + 2 (- B) + 2 (- 6 D) - 2 B - 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.4.2.1.1.2.2

Multiply $- 1$ by $2$ .

$- 14 = 54 - 2 B + 2 (- 6 D) - 2 B - 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.4.2.1.1.2.3

Multiply $- 6$ by $2$ .

$- 14 = 54 - 2 B - 12 D - 2 B - 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 54 - 2 B - 12 D - 2 B - 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 54 - 2 B - 12 D - 2 B - 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.4.2.1.2

Simplify by adding terms.

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Step 3.4.2.1.2.1

Subtract $2 B$ from $- 2 B$ .

$- 14 = 54 - 4 B - 12 D - 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.4.2.1.2.2

Subtract $3 D$ from $- 12 D$ .

$- 14 = 54 - 4 B - 15 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 54 - 4 B - 15 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 54 - 4 B - 15 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 54 - 4 B - 15 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

$- 14 = 54 - 4 B - 15 D$

$A = 27 - B - 6 D$

$C = - D$

$- 10 = B - 3 D$

Step 3.5

Solve for $B$ in $- 10 = B - 3 D$ .

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Step 3.5.1

Rewrite the equation as $B - 3 D = - 10$ .

$B - 3 D = - 10$

$- 14 = 54 - 4 B - 15 D$

$A = 27 - B - 6 D$

$C = - D$

Step 3.5.2

Add $3 D$ to both sides of the equation.

$B = - 10 + 3 D$

$- 14 = 54 - 4 B - 15 D$

$A = 27 - B - 6 D$

$C = - D$

$B = - 10 + 3 D$

$- 14 = 54 - 4 B - 15 D$

$A = 27 - B - 6 D$

$C = - D$

Step 3.6

Replace all occurrences of $B$ with $- 10 + 3 D$ in each equation.

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Step 3.6.1

Replace all occurrences of $B$ in $- 14 = 54 - 4 B - 15 D$ with $- 10 + 3 D$ .

$- 14 = 54 - 4 (- 10 + 3 D) - 15 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $54 - 4 (- 10 + 3 D) - 15 D$ .

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Step 3.6.2.1.1

Simplify each term.

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Step 3.6.2.1.1.1

Apply the distributive property.

$- 14 = 54 - 4 \cdot - 10 - 4 (3 D) - 15 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

Step 3.6.2.1.1.2

Multiply $- 4$ by $- 10$ .

$- 14 = 54 + 40 - 4 (3 D) - 15 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

Step 3.6.2.1.1.3

Multiply $3$ by $- 4$ .

$- 14 = 54 + 40 - 12 D - 15 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 14 = 54 + 40 - 12 D - 15 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

Step 3.6.2.1.2

Simplify by adding terms.

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Step 3.6.2.1.2.1

Add $54$ and $40$ .

$- 14 = 94 - 12 D - 15 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

Step 3.6.2.1.2.2

Subtract $15 D$ from $- 12 D$ .

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$A = 27 - B - 6 D$

$C = - D$

Step 3.6.3

Replace all occurrences of $B$ in $A = 27 - B - 6 D$ with $- 10 + 3 D$ .

$A = 27 - (- 10 + 3 D) - 6 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $27 - (- 10 + 3 D) - 6 D$ .

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Step 3.6.4.1.1

Simplify each term.

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Step 3.6.4.1.1.1

Apply the distributive property.

$A = 27 + 10 - (3 D) - 6 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.6.4.1.1.2

Multiply $- 1$ by $- 10$ .

$A = 27 + 10 - (3 D) - 6 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.6.4.1.1.3

Multiply $3$ by $- 1$ .

$A = 27 + 10 - 3 D - 6 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

$A = 27 + 10 - 3 D - 6 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.6.4.1.2

Simplify by adding terms.

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Step 3.6.4.1.2.1

Add $27$ and $10$ .

$A = 37 - 3 D - 6 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.6.4.1.2.2

Subtract $6 D$ from $- 3 D$ .

$A = 37 - 9 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

$A = 37 - 9 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

$A = 37 - 9 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

$A = 37 - 9 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

$A = 37 - 9 D$

$- 14 = 94 - 27 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.7

Solve for $D$ in $- 14 = 94 - 27 D$ .

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Step 3.7.1

Rewrite the equation as $94 - 27 D = - 14$ .

$94 - 27 D = - 14$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.7.2

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

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Step 3.7.2.1

Subtract $94$ from both sides of the equation.

$- 27 D = - 14 - 94$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.7.2.2

Subtract $94$ from $- 14$ .

$- 27 D = - 108$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

$- 27 D = - 108$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.7.3

Divide each term in $- 27 D = - 108$ by $- 27$ and simplify.

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Step 3.7.3.1

Divide each term in $- 27 D = - 108$ by $- 27$ .

$\frac{- 27 D}{- 27} = \frac{- 108}{- 27}$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.7.3.2

Simplify the left side.

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Step 3.7.3.2.1

Cancel the common factor of $- 27$ .

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Step 3.7.3.2.1.1

Cancel the common factor.

$\frac{- 27 D}{- 27} = \frac{- 108}{- 27}$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.7.3.2.1.2

Divide $D$ by $1$ .

$D = \frac{- 108}{- 27}$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

$D = \frac{- 108}{- 27}$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

$D = \frac{- 108}{- 27}$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.7.3.3

Simplify the right side.

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Step 3.7.3.3.1

Divide $- 108$ by $- 27$ .

$D = 4$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

$D = 4$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

$D = 4$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

$D = 4$

$A = 37 - 9 D$

$B = - 10 + 3 D$

$C = - D$

Step 3.8

Replace all occurrences of $D$ with $4$ in each equation.

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Step 3.8.1

Replace all occurrences of $D$ in $A = 37 - 9 D$ with $4$ .

$A = 37 - 9 \cdot 4$

$D = 4$

$B = - 10 + 3 D$

$C = - D$

Step 3.8.2

Simplify the right side.

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Step 3.8.2.1

Simplify $37 - 9 \cdot 4$ .

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Step 3.8.2.1.1

Multiply $- 9$ by $4$ .

$A = 37 - 36$

$D = 4$

$B = - 10 + 3 D$

$C = - D$

Step 3.8.2.1.2

Subtract $36$ from $37$ .

$A = 1$

$D = 4$

$B = - 10 + 3 D$

$C = - D$

$A = 1$

$D = 4$

$B = - 10 + 3 D$

$C = - D$

$A = 1$

$D = 4$

$B = - 10 + 3 D$

$C = - D$

Step 3.8.3

Replace all occurrences of $D$ in $B = - 10 + 3 D$ with $4$ .

$B = - 10 + 3 (4)$

$A = 1$

$D = 4$

$C = - D$

Step 3.8.4

Simplify the right side.

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Step 3.8.4.1

Simplify $- 10 + 3 (4)$ .

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Step 3.8.4.1.1

Multiply $3$ by $4$ .

$B = - 10 + 12$

$A = 1$

$D = 4$

$C = - D$

Step 3.8.4.1.2

Add $- 10$ and $12$ .

$B = 2$

$A = 1$

$D = 4$

$C = - D$

$B = 2$

$A = 1$

$D = 4$

$C = - D$

$B = 2$

$A = 1$

$D = 4$

$C = - D$

Step 3.8.5

Replace all occurrences of $D$ in $C = - D$ with $4$ .

$C = - (4)$

$B = 2$

$A = 1$

$D = 4$

Step 3.8.6

Simplify the right side.

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Step 3.8.6.1

Multiply $- 1$ by $4$ .

$C = - 4$

$B = 2$

$A = 1$

$D = 4$

$C = - 4$

$B = 2$

$A = 1$

$D = 4$

$C = - 4$

$B = 2$

$A = 1$

$D = 4$

Step 3.9

List all of the solutions.

$C = - 4, B = 2, A = 1, D = 4$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{{(x - 1)}^{3}} + \frac{B}{{(x - 1)}^{2}} + \frac{C}{x - 1} + \frac{D}{x + 2}$ with the values found for $A$ , $B$ , $C$ , and $D$ .

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