Calculus Examples

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

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

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 1.2

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

Step 1.3

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

Step 1.4

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

Step 1.5

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

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

Step 1.6

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

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Step 1.6.1

Cancel the common factor.

Step 1.6.2

Rewrite the expression.

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

Step 1.7

Cancel the common factor of $x - 3$ .

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Step 1.7.1

Cancel the common factor.

Step 1.7.2

Divide $2 x + 3$ by $1$ .

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

Step 1.8

Simplify each term.

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Step 1.8.1

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

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Step 1.8.1.1

Cancel the common factor.

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

Step 1.8.1.2

Divide $(A) (x - 3)$ by $1$ .

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

Step 1.8.2

Apply the distributive property.

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

Step 1.8.3

Move $- 3$ to the left of $A$ .

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

Step 1.8.4

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

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Step 1.8.4.1

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

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

Step 1.8.4.2

Cancel the common factors.

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Step 1.8.4.2.1

Multiply by $1$ .

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

Step 1.8.4.2.2

Cancel the common factor.

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

Step 1.8.4.2.3

Rewrite the expression.

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

Step 1.8.4.2.4

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

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

Step 1.8.5

Apply the distributive property.

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

Step 1.8.6

Multiply $B$ by $1$ .

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

Step 1.8.7

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

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Step 1.8.7.1

Apply the distributive property.

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

Step 1.8.7.2

Apply the distributive property.

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

Step 1.8.7.3

Apply the distributive property.

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

Step 1.8.8

Simplify and combine like terms.

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Step 1.8.8.1

Simplify each term.

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Step 1.8.8.1.1

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

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Step 1.8.8.1.1.1

Move $x$ .

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

Step 1.8.8.1.1.2

Multiply $x$ by $x$ .

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

Step 1.8.8.1.2

Move $- 3$ to the left of $B x$ .

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

Step 1.8.8.1.3

Move $- 3$ to the left of $B$ .

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

Step 1.8.8.2

Add $- 3 B x$ and $B x$ .

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

Step 1.8.9

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

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Step 1.8.9.1

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

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

Step 1.8.9.2

Cancel the common factors.

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Step 1.8.9.2.1

Multiply by $1$ .

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

Step 1.8.9.2.2

Cancel the common factor.

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

Step 1.8.9.2.3

Rewrite the expression.

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

Step 1.8.9.2.4

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

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

Step 1.8.10

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

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

Step 1.8.11

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

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Step 1.8.11.1

Apply the distributive property.

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

Step 1.8.11.2

Apply the distributive property.

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

Step 1.8.11.3

Apply the distributive property.

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

Step 1.8.12

Simplify and combine like terms.

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Step 1.8.12.1

Simplify each term.

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Step 1.8.12.1.1

Multiply $x$ by $x$ .

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

Step 1.8.12.1.2

Multiply $x$ by $1$ .

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

Step 1.8.12.1.3

Multiply $x$ by $1$ .

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

Step 1.8.12.1.4

Multiply $1$ by $1$ .

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

Step 1.8.12.2

Add $x$ and $x$ .

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

Step 1.8.13

Apply the distributive property.

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

Step 1.8.14

Simplify.

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Step 1.8.14.1

Rewrite using the commutative property of multiplication.

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

Step 1.8.14.2

Multiply $C$ by $1$ .

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

Step 1.8.15

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

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

Step 1.8.16

Simplify each term.

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Step 1.8.16.1

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

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Step 1.8.16.1.1

Move $x$ .

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

Step 1.8.16.1.2

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

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Step 1.8.16.1.2.1

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

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

Step 1.8.16.1.2.2

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

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

Step 1.8.16.1.3

Add $1$ and $2$ .

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

Step 1.8.16.2

Move $- 3$ to the left of $C x^{2}$ .

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

Step 1.8.16.3

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

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Step 1.8.16.3.1

Move $x$ .

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

Step 1.8.16.3.2

Multiply $x$ by $x$ .

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

Step 1.8.16.4

Multiply $- 3$ by $2$ .

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

Step 1.8.16.5

Move $- 3$ to the left of $C$ .

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

Step 1.8.17

Add $- 3 C x^{2}$ and $2 C x^{2}$ .

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

Step 1.8.18

Add $- 6 C x$ and $C x$ .

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + \frac{(D) {(x + 1)}^{3} (x - 3)}{x - 3}$

Step 1.8.19

Cancel the common factor of $x - 3$ .

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Step 1.8.19.1

Cancel the common factor.

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + \frac{(D) {(x + 1)}^{3} (x - 3)}{x - 3}$

Step 1.8.19.2

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

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + (D) {(x + 1)}^{3}$

Step 1.8.20

Use the Binomial Theorem.

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3})$

Step 1.8.21

Simplify each term.

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Step 1.8.21.1

Multiply $3$ by $1$ .

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3})$

Step 1.8.21.2

One to any power is one.

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3})$

Step 1.8.21.3

Multiply $3$ by $1$ .

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D (x^{3} + 3 x^{2} + 3 x + 1^{3})$

Step 1.8.21.4

One to any power is one.

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D (x^{3} + 3 x^{2} + 3 x + 1)$

Step 1.8.22

Apply the distributive property.

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D x^{3} + D (3 x^{2}) + D (3 x) + D \cdot 1$

Step 1.8.23

Simplify.

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Step 1.8.23.1

Rewrite using the commutative property of multiplication.

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D x^{3} + 3 D x^{2} + D (3 x) + D \cdot 1$

Step 1.8.23.2

Rewrite using the commutative property of multiplication.

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D x^{3} + 3 D x^{2} + 3 D x + D \cdot 1$

Step 1.8.23.3

Multiply $D$ by $1$ .

$2 x + 3 = A x - 3 A + B x^{2} - 2 B x - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D x^{3} + 3 D x^{2} + 3 D x + D$

Step 1.9

Simplify the expression.

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Step 1.9.1

Move $B$ .

$2 x + 3 = A x - 3 A + B x^{2} - 2 x B - 3 B + C x^{3} - C x^{2} - 5 C x - 3 C + D x^{3} + 3 D x^{2} + 3 D x + D$

Step 1.9.2

Move $- 3 C$ .

$2 x + 3 = A x - 3 A + B x^{2} - 2 x B - 3 B + C x^{3} - C x^{2} - 5 C x + D x^{3} + 3 D x^{2} + 3 D x - 3 C + D$

Step 1.9.3

Move $- 3 B$ .

$2 x + 3 = A x - 3 A + B x^{2} - 2 x B + C x^{3} - C x^{2} - 5 C x + D x^{3} + 3 D x^{2} + 3 D x - 3 B - 3 C + D$

Step 1.9.4

Move $- 3 A$ .

$2 x + 3 = A x + B x^{2} - 2 x B + C x^{3} - C x^{2} - 5 C x + D x^{3} + 3 D x^{2} + 3 D x - 3 A - 3 B - 3 C + D$

Step 1.9.5

Move $- 5 C x$ .

$2 x + 3 = A x + B x^{2} - 2 x B + C x^{3} - C x^{2} + D x^{3} + 3 D x^{2} - 5 C x + 3 D x - 3 A - 3 B - 3 C + D$

Step 1.9.6

Move $- 2 x B$ .

$2 x + 3 = A x + B x^{2} + C x^{3} - C x^{2} + D x^{3} + 3 D x^{2} - 2 x B - 5 C x + 3 D x - 3 A - 3 B - 3 C + D$

Step 1.9.7

Move $A x$ .

$2 x + 3 = B x^{2} + C x^{3} - C x^{2} + D x^{3} + 3 D x^{2} + A x - 2 x B - 5 C x + 3 D x - 3 A - 3 B - 3 C + D$

Step 1.9.8

Move $- C x^{2}$ .

$2 x + 3 = B x^{2} + C x^{3} + D x^{3} - C x^{2} + 3 D x^{2} + A x - 2 x B - 5 C x + 3 D x - 3 A - 3 B - 3 C + D$

Step 1.9.9

Move $B x^{2}$ .

$2 x + 3 = C x^{3} + D x^{3} + B x^{2} - C x^{2} + 3 D x^{2} + A x - 2 x B - 5 C x + 3 D x - 3 A - 3 B - 3 C + D$

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.

$0 = B - 1 C + 3 D$

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.

$2 = A - 2 B - 5 C + 3 D$

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.

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

Step 2.5

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

$0 = C + D$

$0 = B - 1 C + 3 D$

$2 = A - 2 B - 5 C + 3 D$

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

$0 = C + D$

$0 = B - 1 C + 3 D$

$2 = A - 2 B - 5 C + 3 D$

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

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 $C + D = 0$ .

$C + D = 0$

$0 = B - 1 C + 3 D$

$2 = A - 2 B - 5 C + 3 D$

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

Step 3.1.2

Subtract $D$ from both sides of the equation.

$C = - D$

$0 = B - 1 C + 3 D$

$2 = A - 2 B - 5 C + 3 D$

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

$C = - D$

$0 = B - 1 C + 3 D$

$2 = A - 2 B - 5 C + 3 D$

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

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 $0 = B - 1 C + 3 D$ with $- D$ .

$0 = B - 1 (- D) + 3 D$

$C = - D$

$2 = A - 2 B - 5 C + 3 D$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $B - 1 (- D) + 3 D$ .

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Step 3.2.2.1.1

Multiply $- 1 (- D)$ .

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Step 3.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$0 = B + 1 D + 3 D$

$C = - D$

$2 = A - 2 B - 5 C + 3 D$

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

Step 3.2.2.1.1.2

Multiply $D$ by $1$ .

$0 = B + D + 3 D$

$C = - D$

$2 = A - 2 B - 5 C + 3 D$

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

$0 = B + D + 3 D$

$C = - D$

$2 = A - 2 B - 5 C + 3 D$

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

Step 3.2.2.1.2

Add $D$ and $3 D$ .

$0 = B + 4 D$

$C = - D$

$2 = A - 2 B - 5 C + 3 D$

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

$0 = B + 4 D$

$C = - D$

$2 = A - 2 B - 5 C + 3 D$

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

$0 = B + 4 D$

$C = - D$

$2 = A - 2 B - 5 C + 3 D$

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

Step 3.2.3

Replace all occurrences of $C$ in $2 = A - 2 B - 5 C + 3 D$ with $- D$ .

$2 = A - 2 B - 5 (- D) + 3 D$

$0 = B + 4 D$

$C = - D$

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

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify $A - 2 B - 5 (- D) + 3 D$ .

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Step 3.2.4.1.1

Multiply $- 1$ by $- 5$ .

$2 = A - 2 B + 5 D + 3 D$

$0 = B + 4 D$

$C = - D$

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

Step 3.2.4.1.2

Add $5 D$ and $3 D$ .

$2 = A - 2 B + 8 D$

$0 = B + 4 D$

$C = - D$

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

$2 = A - 2 B + 8 D$

$0 = B + 4 D$

$C = - D$

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

$2 = A - 2 B + 8 D$

$0 = B + 4 D$

$C = - D$

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

Step 3.2.5

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

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

$2 = A - 2 B + 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.2.6

Simplify the right side.

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Step 3.2.6.1

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

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Step 3.2.6.1.1

Multiply $- 1$ by $- 3$ .

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

$2 = A - 2 B + 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.2.6.1.2

Add $3 D$ and $D$ .

$3 = - 3 A - 3 B + 4 D$

$2 = A - 2 B + 8 D$

$0 = B + 4 D$

$C = - D$

$3 = - 3 A - 3 B + 4 D$

$2 = A - 2 B + 8 D$

$0 = B + 4 D$

$C = - D$

$3 = - 3 A - 3 B + 4 D$

$2 = A - 2 B + 8 D$

$0 = B + 4 D$

$C = - D$

$3 = - 3 A - 3 B + 4 D$

$2 = A - 2 B + 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.3

Solve for $A$ in $2 = A - 2 B + 8 D$ .

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Step 3.3.1

Rewrite the equation as $A - 2 B + 8 D = 2$ .

$A - 2 B + 8 D = 2$

$3 = - 3 A - 3 B + 4 D$

$0 = B + 4 D$

$C = - D$

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

Add $2 B$ to both sides of the equation.

$A + 8 D = 2 + 2 B$

$3 = - 3 A - 3 B + 4 D$

$0 = B + 4 D$

$C = - D$

Step 3.3.2.2

Subtract $8 D$ from both sides of the equation.

$A = 2 + 2 B - 8 D$

$3 = - 3 A - 3 B + 4 D$

$0 = B + 4 D$

$C = - D$

$A = 2 + 2 B - 8 D$

$3 = - 3 A - 3 B + 4 D$

$0 = B + 4 D$

$C = - D$

$A = 2 + 2 B - 8 D$

$3 = - 3 A - 3 B + 4 D$

$0 = B + 4 D$

$C = - D$

Step 3.4

Replace all occurrences of $A$ with $2 + 2 B - 8 D$ in each equation.

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Step 3.4.1

Replace all occurrences of $A$ in $3 = - 3 A - 3 B + 4 D$ with $2 + 2 B - 8 D$ .

$3 = - 3 (2 + 2 B - 8 D) - 3 B + 4 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 3 (2 + 2 B - 8 D) - 3 B + 4 D$ .

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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.

$3 = - 3 \cdot 2 - 3 (2 B) - 3 (- 8 D) - 3 B + 4 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.4.2.1.1.2

Simplify.

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Step 3.4.2.1.1.2.1

Multiply $- 3$ by $2$ .

$3 = - 6 - 3 (2 B) - 3 (- 8 D) - 3 B + 4 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.4.2.1.1.2.2

Multiply $2$ by $- 3$ .

$3 = - 6 - 6 B - 3 (- 8 D) - 3 B + 4 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.4.2.1.1.2.3

Multiply $- 8$ by $- 3$ .

$3 = - 6 - 6 B + 24 D - 3 B + 4 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

$3 = - 6 - 6 B + 24 D - 3 B + 4 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

$3 = - 6 - 6 B + 24 D - 3 B + 4 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.4.2.1.2

Simplify by adding terms.

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Step 3.4.2.1.2.1

Subtract $3 B$ from $- 6 B$ .

$3 = - 6 - 9 B + 24 D + 4 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.4.2.1.2.2

Add $24 D$ and $4 D$ .

$3 = - 6 - 9 B + 28 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

$3 = - 6 - 9 B + 28 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

$3 = - 6 - 9 B + 28 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

$3 = - 6 - 9 B + 28 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

$3 = - 6 - 9 B + 28 D$

$A = 2 + 2 B - 8 D$

$0 = B + 4 D$

$C = - D$

Step 3.5

Solve for $B$ in $0 = B + 4 D$ .

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Step 3.5.1

Rewrite the equation as $B + 4 D = 0$ .

$B + 4 D = 0$

$3 = - 6 - 9 B + 28 D$

$A = 2 + 2 B - 8 D$

$C = - D$

Step 3.5.2

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

$B = - 4 D$

$3 = - 6 - 9 B + 28 D$

$A = 2 + 2 B - 8 D$

$C = - D$

$B = - 4 D$

$3 = - 6 - 9 B + 28 D$

$A = 2 + 2 B - 8 D$

$C = - D$

Step 3.6

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

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Step 3.6.1

Replace all occurrences of $B$ in $3 = - 6 - 9 B + 28 D$ with $- 4 D$ .

$3 = - 6 - 9 (- 4 D) + 28 D$

$B = - 4 D$

$A = 2 + 2 B - 8 D$

$C = - D$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $- 6 - 9 (- 4 D) + 28 D$ .

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Step 3.6.2.1.1

Multiply $- 4$ by $- 9$ .

$3 = - 6 + 36 D + 28 D$

$B = - 4 D$

$A = 2 + 2 B - 8 D$

$C = - D$

Step 3.6.2.1.2

Add $36 D$ and $28 D$ .

$3 = - 6 + 64 D$

$B = - 4 D$

$A = 2 + 2 B - 8 D$

$C = - D$

$3 = - 6 + 64 D$

$B = - 4 D$

$A = 2 + 2 B - 8 D$

$C = - D$

$3 = - 6 + 64 D$

$B = - 4 D$

$A = 2 + 2 B - 8 D$

$C = - D$

Step 3.6.3

Replace all occurrences of $B$ in $A = 2 + 2 B - 8 D$ with $- 4 D$ .

$A = 2 + 2 (- 4 D) - 8 D$

$3 = - 6 + 64 D$

$B = - 4 D$

$C = - D$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $2 + 2 (- 4 D) - 8 D$ .

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Step 3.6.4.1.1

Multiply $- 4$ by $2$ .

$A = 2 - 8 D - 8 D$

$3 = - 6 + 64 D$

$B = - 4 D$

$C = - D$

Step 3.6.4.1.2

Subtract $8 D$ from $- 8 D$ .

$A = 2 - 16 D$

$3 = - 6 + 64 D$

$B = - 4 D$

$C = - D$

$A = 2 - 16 D$

$3 = - 6 + 64 D$

$B = - 4 D$

$C = - D$

$A = 2 - 16 D$

$3 = - 6 + 64 D$

$B = - 4 D$

$C = - D$

$A = 2 - 16 D$

$3 = - 6 + 64 D$

$B = - 4 D$

$C = - D$

Step 3.7

Solve for $D$ in $3 = - 6 + 64 D$ .

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Step 3.7.1

Rewrite the equation as $- 6 + 64 D = 3$ .

$- 6 + 64 D = 3$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

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

Add $6$ to both sides of the equation.

$64 D = 3 + 6$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

Step 3.7.2.2

Add $3$ and $6$ .

$64 D = 9$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

$64 D = 9$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

Step 3.7.3

Divide each term in $64 D = 9$ by $64$ and simplify.

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Step 3.7.3.1

Divide each term in $64 D = 9$ by $64$ .

$\frac{64 D}{64} = \frac{9}{64}$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

Step 3.7.3.2

Simplify the left side.

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Step 3.7.3.2.1

Cancel the common factor of $64$ .

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Step 3.7.3.2.1.1

Cancel the common factor.

$\frac{64 D}{64} = \frac{9}{64}$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

Step 3.7.3.2.1.2

Divide $D$ by $1$ .

$D = \frac{9}{64}$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

$D = \frac{9}{64}$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

$D = \frac{9}{64}$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

$D = \frac{9}{64}$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

$D = \frac{9}{64}$

$A = 2 - 16 D$

$B = - 4 D$

$C = - D$

Step 3.8

Replace all occurrences of $D$ with $\frac{9}{64}$ in each equation.

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Step 3.8.1

Replace all occurrences of $D$ in $A = 2 - 16 D$ with $\frac{9}{64}$ .

$A = 2 - 16 (\frac{9}{64})$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2

Simplify the right side.

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Step 3.8.2.1

Simplify $2 - 16 (\frac{9}{64})$ .

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Step 3.8.2.1.1

Simplify each term.

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Step 3.8.2.1.1.1

Cancel the common factor of $16$ .

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Step 3.8.2.1.1.1.1

Factor $16$ out of $- 16$ .

$A = 2 + 16 (- 1) (\frac{9}{64})$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.1.1.2

Factor $16$ out of $64$ .

$A = 2 + 16 \cdot (- 1 \frac{9}{16 \cdot 4})$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.1.1.3

Cancel the common factor.

$A = 2 + 16 \cdot (- 1 \frac{9}{16 \cdot 4})$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.1.1.4

Rewrite the expression.

$A = 2 - 1 (\frac{9}{4})$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

$A = 2 - 1 (\frac{9}{4})$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.1.2

Rewrite $- 1 (\frac{9}{4})$ as $- (\frac{9}{4})$ .

$A = 2 - \frac{9}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

$A = 2 - \frac{9}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$A = 2 \cdot \frac{4}{4} - \frac{9}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.3

Combine $2$ and $\frac{4}{4}$ .

$A = \frac{2 \cdot 4}{4} - \frac{9}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.4

Combine the numerators over the common denominator.

$A = \frac{2 \cdot 4 - 9}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.5

Simplify the numerator.

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Step 3.8.2.1.5.1

Multiply $2$ by $4$ .

$A = \frac{8 - 9}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.5.2

Subtract $9$ from $8$ .

$A = \frac{- 1}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

$A = \frac{- 1}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.2.1.6

Move the negative in front of the fraction.

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$B = - 4 D$

$C = - D$

Step 3.8.3

Replace all occurrences of $D$ in $B = - 4 D$ with $\frac{9}{64}$ .

$B = - 4 (\frac{9}{64})$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - D$

Step 3.8.4

Simplify the right side.

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Step 3.8.4.1

Simplify $- 4 (\frac{9}{64})$ .

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Step 3.8.4.1.1

Cancel the common factor of $4$ .

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Step 3.8.4.1.1.1

Factor $4$ out of $- 4$ .

$B = 4 (- 1) (\frac{9}{64})$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - D$

Step 3.8.4.1.1.2

Factor $4$ out of $64$ .

$B = 4 \cdot (- 1 \frac{9}{4 \cdot 16})$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - D$

Step 3.8.4.1.1.3

Cancel the common factor.

$B = 4 \cdot (- 1 \frac{9}{4 \cdot 16})$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - D$

Step 3.8.4.1.1.4

Rewrite the expression.

$B = - 1 (\frac{9}{16})$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - D$

$B = - 1 (\frac{9}{16})$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - D$

Step 3.8.4.1.2

Rewrite $- 1 (\frac{9}{16})$ as $- (\frac{9}{16})$ .

$B = - \frac{9}{16}$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - D$

$B = - \frac{9}{16}$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - D$

$B = - \frac{9}{16}$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - D$

Step 3.8.5

Replace all occurrences of $D$ in $C = - D$ with $\frac{9}{64}$ .

$C = - (\frac{9}{64})$

$B = - \frac{9}{16}$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

Step 3.8.6

Simplify the right side.

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Step 3.8.6.1

Multiply $- 1$ by $\frac{9}{64}$ .

$C = - \frac{9}{64}$

$B = - \frac{9}{16}$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - \frac{9}{64}$

$B = - \frac{9}{16}$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

$C = - \frac{9}{64}$

$B = - \frac{9}{16}$

$A = - \frac{1}{4}$

$D = \frac{9}{64}$

Step 3.9

List all of the solutions.

$C = - \frac{9}{64}, B = - \frac{9}{16}, A = - \frac{1}{4}, D = \frac{9}{64}$

Step 4

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

$\frac{- \frac{1}{4}}{{(x + 1)}^{3}} + \frac{- \frac{9}{16}}{{(x + 1)}^{2}} + \frac{- \frac{9}{64}}{x + 1} + \frac{\frac{9}{64}}{x - 3}$