Precalculus Examples

$\frac{3 x^{2} + 7 x - 10}{x^{3} + 2 x^{2} - 4 x - 8}$

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

Factor by grouping.

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Step 1.1.1.1

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 $a \cdot c = 3 \cdot - 10 = - 30$ and whose sum is $b = 7$ .

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Step 1.1.1.1.1

Factor $7$ out of $7 x$ .

$\frac{3 x^{2} + 7 (x) - 10}{x^{3} + 2 x^{2} - 4 x - 8}$

Step 1.1.1.1.2

Rewrite $7$ as $- 3$ plus $10$

$\frac{3 x^{2} + (- 3 + 10) x - 10}{x^{3} + 2 x^{2} - 4 x - 8}$

Step 1.1.1.1.3

Apply the distributive property.

$\frac{3 x^{2} - 3 x + 10 x - 10}{x^{3} + 2 x^{2} - 4 x - 8}$

Step 1.1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.1.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} - 3 x) + 10 x - 10}{x^{3} + 2 x^{2} - 4 x - 8}$

Step 1.1.1.2.2

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

$\frac{3 x (x - 1) + 10 (x - 1)}{x^{3} + 2 x^{2} - 4 x - 8}$

Step 1.1.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$\frac{(x - 1) (3 x + 10)}{x^{3} + 2 x^{2} - 4 x - 8}$

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.

$\frac{(x - 1) (3 x + 10)}{(x^{3} + 2 x^{2}) - 4 x - 8}$

Step 1.1.2.2

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

$\frac{(x - 1) (3 x + 10)}{x^{2} (x + 2) - 4 (x + 2)}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$\frac{(x - 1) (3 x + 10)}{(x + 2) (x^{2} - 4)}$

Step 1.1.4

Rewrite $4$ as $2^{2}$ .

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

Step 1.1.5

Factor.

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Step 1.1.5.1

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

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

Step 1.1.5.2

Remove unnecessary parentheses.

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

Step 1.1.6

Combine exponents.

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Step 1.1.6.1

Raise $x + 2$ to the power of $1$ .

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

Step 1.1.6.2

Raise $x + 2$ to the power of $1$ .

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

Step 1.1.6.3

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

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

Step 1.1.6.4

Add $1$ and $1$ .

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

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 $A$ .

$\frac{A}{{(x + 2)}^{2}}$

Step 1.3

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

Step 1.4

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

Step 1.5

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

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

Step 1.6

Reduce the expression by cancelling the common factors.

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Step 1.6.1

Cancel the common factor of ${(x + 2)}^{2}$ .

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Step 1.6.1.1

Cancel the common factor.

Step 1.6.1.2

Rewrite the expression.

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

Step 1.6.2

Cancel the common factor of $x - 2$ .

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Step 1.6.2.1

Cancel the common factor.

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

Step 1.6.2.2

Divide $(x - 1) (3 x + 10)$ by $1$ .

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

Step 1.7

Expand $(x - 1) (3 x + 10)$ using the FOIL Method.

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Step 1.7.1

Apply the distributive property.

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

Apply the distributive property.

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

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.8.1.2

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

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Step 1.8.1.2.1

Move $x$ .

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

Step 1.8.1.2.2

Multiply $x$ by $x$ .

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

Step 1.8.1.3

Move $10$ to the left of $x$ .

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

Step 1.8.1.4

Multiply $3$ by $- 1$ .

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

Step 1.8.1.5

Multiply $- 1$ by $10$ .

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

Step 1.8.2

Subtract $3 x$ from $10 x$ .

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

Step 1.9

Simplify each term.

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Step 1.9.1

Cancel the common factor of ${(x + 2)}^{2}$ .

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Step 1.9.1.1

Cancel the common factor.

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

Step 1.9.1.2

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

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

Step 1.9.2

Apply the distributive property.

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

Step 1.9.3

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

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

Step 1.9.4

Cancel the common factor of ${(x + 2)}^{2}$ and $x + 2$ .

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Step 1.9.4.1

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

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

Step 1.9.4.2

Cancel the common factors.

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Step 1.9.4.2.1

Multiply by $1$ .

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

Step 1.9.4.2.2

Cancel the common factor.

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

Step 1.9.4.2.3

Rewrite the expression.

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

Step 1.9.4.2.4

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

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

Step 1.9.5

Apply the distributive property.

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

Step 1.9.6

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

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

Step 1.9.7

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

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Step 1.9.7.1

Apply the distributive property.

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

Step 1.9.7.2

Apply the distributive property.

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

Step 1.9.7.3

Apply the distributive property.

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

Step 1.9.8

Combine the opposite terms in $B x \cdot x + B x \cdot - 2 + 2 B x + 2 B \cdot - 2$ .

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Step 1.9.8.1

Reorder the factors in the terms $B x \cdot - 2$ and $2 B x$ .

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

Step 1.9.8.2

Add $- 2 B x$ and $2 B x$ .

$3 x^{2} + 7 x - 10 = A x - 2 A + B x \cdot x + 0 + 2 B \cdot - 2 + \frac{(C) {(x + 2)}^{2} (x - 2)}{x - 2}$

Step 1.9.8.3

Add $B x \cdot x$ and $0$ .

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

Step 1.9.9

Simplify each term.

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Step 1.9.9.1

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

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Step 1.9.9.1.1

Move $x$ .

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

Step 1.9.9.1.2

Multiply $x$ by $x$ .

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

Step 1.9.9.2

Multiply $- 2$ by $2$ .

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

Step 1.9.10

Cancel the common factor of $x - 2$ .

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Step 1.9.10.1

Cancel the common factor.

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

Step 1.9.10.2

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

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + (C) {(x + 2)}^{2}$

Step 1.9.11

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C ((x + 2) (x + 2))$

Step 1.9.12

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

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Step 1.9.12.1

Apply the distributive property.

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C (x (x + 2) + 2 (x + 2))$

Step 1.9.12.2

Apply the distributive property.

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C (x \cdot x + x \cdot 2 + 2 (x + 2))$

Step 1.9.12.3

Apply the distributive property.

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2)$

Step 1.9.13

Simplify and combine like terms.

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Step 1.9.13.1

Simplify each term.

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Step 1.9.13.1.1

Multiply $x$ by $x$ .

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2)$

Step 1.9.13.1.2

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

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2)$

Step 1.9.13.1.3

Multiply $2$ by $2$ .

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C (x^{2} + 2 x + 2 x + 4)$

Step 1.9.13.2

Add $2 x$ and $2 x$ .

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C (x^{2} + 4 x + 4)$

Step 1.9.14

Apply the distributive property.

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C x^{2} + C (4 x) + C \cdot 4$

Step 1.9.15

Simplify.

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Step 1.9.15.1

Rewrite using the commutative property of multiplication.

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C x^{2} + 4 C x + C \cdot 4$

Step 1.9.15.2

Move $4$ to the left of $C$ .

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C x^{2} + 4 C x + 4 \cdot C$

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} - 4 B + C x^{2} + 4 C x + 4 C$

Step 1.10

Simplify the expression.

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Step 1.10.1

Move $- 4 B$ .

$3 x^{2} + 7 x - 10 = A x - 2 A + B x^{2} + C x^{2} + 4 C x - 4 B + 4 C$

Step 1.10.2

Move $- 2 A$ .

$3 x^{2} + 7 x - 10 = A x + B x^{2} + C x^{2} + 4 C x - 2 A - 4 B + 4 C$

Step 1.10.3

Move $A x$ .

$3 x^{2} + 7 x - 10 = B x^{2} + C x^{2} + A x + 4 C x - 2 A - 4 B + 4 C$

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.

$3 = B + C$

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.

$7 = A + 4 C$

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.

$- 10 = - 2 A - 4 B + 4 C$

Step 2.4

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

$3 = B + C$

$7 = A + 4 C$

$- 10 = - 2 A - 4 B + 4 C$

$3 = B + C$

$7 = A + 4 C$

$- 10 = - 2 A - 4 B + 4 C$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $B$ in $3 = B + C$ .

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Step 3.1.1

Rewrite the equation as $B + C = 3$ .

$B + C = 3$

$7 = A + 4 C$

$- 10 = - 2 A - 4 B + 4 C$

Step 3.1.2

Subtract $C$ from both sides of the equation.

$B = 3 - C$

$7 = A + 4 C$

$- 10 = - 2 A - 4 B + 4 C$

$B = 3 - C$

$7 = A + 4 C$

$- 10 = - 2 A - 4 B + 4 C$

Step 3.2

Replace all occurrences of $B$ with $3 - C$ in each equation.

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Step 3.2.1

Replace all occurrences of $B$ in $- 10 = - 2 A - 4 B + 4 C$ with $3 - C$ .

$- 10 = - 2 A - 4 (3 - C) + 4 C$

$B = 3 - C$

$7 = A + 4 C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 2 A - 4 (3 - C) + 4 C$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$- 10 = - 2 A - 4 \cdot 3 - 4 (- C) + 4 C$

$B = 3 - C$

$7 = A + 4 C$

Step 3.2.2.1.1.2

Multiply $- 4$ by $3$ .

$- 10 = - 2 A - 12 - 4 (- C) + 4 C$

$B = 3 - C$

$7 = A + 4 C$

Step 3.2.2.1.1.3

Multiply $- 1$ by $- 4$ .

$- 10 = - 2 A - 12 + 4 C + 4 C$

$B = 3 - C$

$7 = A + 4 C$

$- 10 = - 2 A - 12 + 4 C + 4 C$

$B = 3 - C$

$7 = A + 4 C$

Step 3.2.2.1.2

Add $4 C$ and $4 C$ .

$- 10 = - 2 A - 12 + 8 C$

$B = 3 - C$

$7 = A + 4 C$

$- 10 = - 2 A - 12 + 8 C$

$B = 3 - C$

$7 = A + 4 C$

$- 10 = - 2 A - 12 + 8 C$

$B = 3 - C$

$7 = A + 4 C$

$- 10 = - 2 A - 12 + 8 C$

$B = 3 - C$

$7 = A + 4 C$

Step 3.3

Reorder $3$ and $- C$ .

$B = - C + 3$

$- 10 = - 2 A - 12 + 8 C$

$7 = A + 4 C$

Step 3.4

Solve for $A$ in $7 = A + 4 C$ .

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Step 3.4.1

Rewrite the equation as $A + 4 C = 7$ .

$A + 4 C = 7$

$B = - C + 3$

$- 10 = - 2 A - 12 + 8 C$

Step 3.4.2

Subtract $4 C$ from both sides of the equation.

$A = 7 - 4 C$

$B = - C + 3$

$- 10 = - 2 A - 12 + 8 C$

$A = 7 - 4 C$

$B = - C + 3$

$- 10 = - 2 A - 12 + 8 C$

Step 3.5

Replace all occurrences of $A$ with $7 - 4 C$ in each equation.

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Step 3.5.1

Replace all occurrences of $A$ in $- 10 = - 2 A - 12 + 8 C$ with $7 - 4 C$ .

$- 10 = - 2 (7 - 4 C) - 12 + 8 C$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $- 2 (7 - 4 C) - 12 + 8 C$ .

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Step 3.5.2.1.1

Simplify each term.

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Step 3.5.2.1.1.1

Apply the distributive property.

$- 10 = - 2 \cdot 7 - 2 (- 4 C) - 12 + 8 C$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.5.2.1.1.2

Multiply $- 2$ by $7$ .

$- 10 = - 14 - 2 (- 4 C) - 12 + 8 C$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.5.2.1.1.3

Multiply $- 4$ by $- 2$ .

$- 10 = - 14 + 8 C - 12 + 8 C$

$A = 7 - 4 C$

$B = - C + 3$

$- 10 = - 14 + 8 C - 12 + 8 C$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.5.2.1.2

Simplify by adding terms.

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Step 3.5.2.1.2.1

Subtract $12$ from $- 14$ .

$- 10 = 8 C - 26 + 8 C$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.5.2.1.2.2

Add $8 C$ and $8 C$ .

$- 10 = 16 C - 26$

$A = 7 - 4 C$

$B = - C + 3$

$- 10 = 16 C - 26$

$A = 7 - 4 C$

$B = - C + 3$

$- 10 = 16 C - 26$

$A = 7 - 4 C$

$B = - C + 3$

$- 10 = 16 C - 26$

$A = 7 - 4 C$

$B = - C + 3$

$- 10 = 16 C - 26$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.6

Solve for $C$ in $- 10 = 16 C - 26$ .

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Step 3.6.1

Rewrite the equation as $16 C - 26 = - 10$ .

$16 C - 26 = - 10$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.6.2

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

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Step 3.6.2.1

Add $26$ to both sides of the equation.

$16 C = - 10 + 26$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.6.2.2

Add $- 10$ and $26$ .

$16 C = 16$

$A = 7 - 4 C$

$B = - C + 3$

$16 C = 16$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.6.3

Divide each term in $16 C = 16$ by $16$ and simplify.

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Step 3.6.3.1

Divide each term in $16 C = 16$ by $16$ .

$\frac{16 C}{16} = \frac{16}{16}$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Cancel the common factor of $16$ .

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Step 3.6.3.2.1.1

Cancel the common factor.

$\frac{16 C}{16} = \frac{16}{16}$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.6.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{16}{16}$

$A = 7 - 4 C$

$B = - C + 3$

$C = \frac{16}{16}$

$A = 7 - 4 C$

$B = - C + 3$

$C = \frac{16}{16}$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Divide $16$ by $16$ .

$C = 1$

$A = 7 - 4 C$

$B = - C + 3$

$C = 1$

$A = 7 - 4 C$

$B = - C + 3$

$C = 1$

$A = 7 - 4 C$

$B = - C + 3$

$C = 1$

$A = 7 - 4 C$

$B = - C + 3$

Step 3.7

Replace all occurrences of $C$ with $1$ in each equation.

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Step 3.7.1

Replace all occurrences of $C$ in $A = 7 - 4 C$ with $1$ .

$A = 7 - 4 \cdot 1$

$C = 1$

$B = - C + 3$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify $7 - 4 \cdot 1$ .

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Step 3.7.2.1.1

Multiply $- 4$ by $1$ .

$A = 7 - 4$

$C = 1$

$B = - C + 3$

Step 3.7.2.1.2

Subtract $4$ from $7$ .

$A = 3$

$C = 1$

$B = - C + 3$

$A = 3$

$C = 1$

$B = - C + 3$

$A = 3$

$C = 1$

$B = - C + 3$

Step 3.7.3

Replace all occurrences of $C$ in $B = - C + 3$ with $1$ .

$B = - (1) + 3$

$A = 3$

$C = 1$

Step 3.7.4

Simplify the right side.

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Step 3.7.4.1

Simplify $- (1) + 3$ .

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Step 3.7.4.1.1

Multiply $- 1$ by $1$ .

$B = - 1 + 3$

$A = 3$

$C = 1$

Step 3.7.4.1.2

Add $- 1$ and $3$ .

$B = 2$

$A = 3$

$C = 1$

$B = 2$

$A = 3$

$C = 1$

$B = 2$

$A = 3$

$C = 1$

$B = 2$

$A = 3$

$C = 1$

Step 3.8

List all of the solutions.

$B = 2, A = 3, C = 1$

Step 4

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

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