Precalculus Examples

$\frac{x^{2} - 3 x + 2}{9 x^{3} + 7 x^{2}}$

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 $x^{2} - 3 x + 2$ using the AC method.

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Step 1.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 1.1.1.2

Write the factored form using these integers.

$\frac{(x - 2) (x - 1)}{9 x^{3} + 7 x^{2}}$

Step 1.1.2

Factor $x^{2}$ out of $9 x^{3} + 7 x^{2}$ .

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Step 1.1.2.1

Factor $x^{2}$ out of $9 x^{3}$ .

$\frac{(x - 2) (x - 1)}{x^{2} (9 x) + 7 x^{2}}$

Step 1.1.2.2

Factor $x^{2}$ out of $7 x^{2}$ .

$\frac{(x - 2) (x - 1)}{x^{2} (9 x) + x^{2} \cdot 7}$

Step 1.1.2.3

Factor $x^{2}$ out of $x^{2} (9 x) + x^{2} \cdot 7$ .

$\frac{(x - 2) (x - 1)}{x^{2} (9 x + 7)}$

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 $C$ .

$\frac{A}{x^{2}} + \frac{B}{x} + \frac{C}{9 x + 7}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $x^{2} (9 x + 7)$ .

$\frac{(x - 2) (x - 1) (x^{2} (9 x + 7))}{x^{2} (9 x + 7)} = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.4

Reduce the expression by cancelling the common factors.

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Step 1.4.1

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

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Step 1.4.1.1

Cancel the common factor.

$\frac{(x - 2) (x - 1) (x^{2} (9 x + 7))}{x^{2} (9 x + 7)} = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.4.1.2

Rewrite the expression.

$\frac{(x - 2) (x - 1) (9 x + 7)}{9 x + 7} = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.4.2

Cancel the common factor of $9 x + 7$ .

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Step 1.4.2.1

Cancel the common factor.

$\frac{(x - 2) (x - 1) (9 x + 7)}{9 x + 7} = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.4.2.2

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

$(x - 2) (x - 1) = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.5

Expand $(x - 2) (x - 1)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$x (x - 1) - 2 (x - 1) = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.5.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 2 (x - 1) = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.5.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 2 x - 2 \cdot - 1 = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 2 x - 2 \cdot - 1 = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.6.1.2

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

$x^{2} - 1 \cdot x - 2 x - 2 \cdot - 1 = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.6.1.3

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

$x^{2} - x - 2 x - 2 \cdot - 1 = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.6.1.4

Multiply $- 2$ by $- 1$ .

$x^{2} - x - 2 x + 2 = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.6.2

Subtract $2 x$ from $- x$ .

$x^{2} - 3 x + 2 = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7

Simplify each term.

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Step 1.7.1

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

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Step 1.7.1.1

Cancel the common factor.

$x^{2} - 3 x + 2 = \frac{A (x^{2} (9 x + 7))}{x^{2}} + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.1.2

Divide $A (9 x + 7)$ by $1$ .

$x^{2} - 3 x + 2 = A (9 x + 7) + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.2

Apply the distributive property.

$x^{2} - 3 x + 2 = A (9 x) + A \cdot 7 + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.3

Rewrite using the commutative property of multiplication.

$x^{2} - 3 x + 2 = 9 A x + A \cdot 7 + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.4

Move $7$ to the left of $A$ .

$x^{2} - 3 x + 2 = 9 A x + 7 \cdot A + \frac{B (x^{2} (9 x + 7))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.5

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

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Step 1.7.5.1

Factor $x$ out of $B (x^{2} (9 x + 7))$ .

$x^{2} - 3 x + 2 = 9 A x + 7 A + \frac{x (B (x (9 x + 7)))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.5.2

Cancel the common factors.

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Step 1.7.5.2.1

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

$x^{2} - 3 x + 2 = 9 A x + 7 A + \frac{x (B (x (9 x + 7)))}{x} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.5.2.2

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

$x^{2} - 3 x + 2 = 9 A x + 7 A + \frac{x (B (x (9 x + 7)))}{x \cdot 1} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.5.2.3

Cancel the common factor.

$x^{2} - 3 x + 2 = 9 A x + 7 A + \frac{x (B (x (9 x + 7)))}{x \cdot 1} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.5.2.4

Rewrite the expression.

$x^{2} - 3 x + 2 = 9 A x + 7 A + \frac{B (x (9 x + 7))}{1} + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.5.2.5

Divide $B (x (9 x + 7))$ by $1$ .

$x^{2} - 3 x + 2 = 9 A x + 7 A + B (x (9 x + 7)) + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.6

Apply the distributive property.

$x^{2} - 3 x + 2 = 9 A x + 7 A + B (x (9 x) + x \cdot 7) + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.7

Rewrite using the commutative property of multiplication.

$x^{2} - 3 x + 2 = 9 A x + 7 A + B (9 x \cdot x + x \cdot 7) + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.8

Move $7$ to the left of $x$ .

$x^{2} - 3 x + 2 = 9 A x + 7 A + B (9 x \cdot x + 7 \cdot x) + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.9

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

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Step 1.7.9.1

Move $x$ .

$x^{2} - 3 x + 2 = 9 A x + 7 A + B (9 (x \cdot x) + 7 \cdot x) + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.9.2

Multiply $x$ by $x$ .

$x^{2} - 3 x + 2 = 9 A x + 7 A + B (9 x^{2} + 7 \cdot x) + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

$x^{2} - 3 x + 2 = 9 A x + 7 A + B (9 x^{2} + 7 x) + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.10

Apply the distributive property.

$x^{2} - 3 x + 2 = 9 A x + 7 A + B (9 x^{2}) + B (7 x) + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.11

Rewrite using the commutative property of multiplication.

$x^{2} - 3 x + 2 = 9 A x + 7 A + 9 B x^{2} + B (7 x) + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.12

Rewrite using the commutative property of multiplication.

$x^{2} - 3 x + 2 = 9 A x + 7 A + 9 B x^{2} + 7 B x + \frac{(C) (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.13

Cancel the common factor of $9 x + 7$ .

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Step 1.7.13.1

Cancel the common factor.

$x^{2} - 3 x + 2 = 9 A x + 7 A + 9 B x^{2} + 7 B x + \frac{C (x^{2} (9 x + 7))}{9 x + 7}$

Step 1.7.13.2

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

$x^{2} - 3 x + 2 = 9 A x + 7 A + 9 B x^{2} + 7 B x + (C) (x^{2})$

$x^{2} - 3 x + 2 = 9 A x + 7 A + 9 B x^{2} + 7 B x + C x^{2}$

Step 1.8

Reorder.

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Step 1.8.1

Move $A$ .

$x^{2} - 3 x + 2 = 9 x A + 7 A + 9 B x^{2} + 7 B x + C x^{2}$

Step 1.8.2

Move $B$ .

$x^{2} - 3 x + 2 = 9 x A + 7 A + 9 x^{2} B + 7 B x + C x^{2}$

Step 1.8.3

Move $B$ .

$x^{2} - 3 x + 2 = 9 x A + 7 A + 9 x^{2} B + 7 x B + C x^{2}$

Step 1.8.4

Move $7 A$ .

$x^{2} - 3 x + 2 = 9 x A + 9 x^{2} B + 7 x B + C x^{2} + 7 A$

Step 1.8.5

Move $7 x B$ .

$x^{2} - 3 x + 2 = 9 x A + 9 x^{2} B + C x^{2} + 7 x B + 7 A$

Step 1.8.6

Move $9 x A$ .

$x^{2} - 3 x + 2 = 9 x^{2} B + C x^{2} + 9 x A + 7 x B + 7 A$

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.

$1 = 9 B + C$

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.

$- 3 = 9 A + 7 B$

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.

$2 = 7 A$

Step 2.4

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

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

$2 = 7 A$

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

$2 = 7 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $2 = 7 A$ .

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Step 3.1.1

Rewrite the equation as $7 A = 2$ .

$7 A = 2$

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

Step 3.1.2

Divide each term in $7 A = 2$ by $7$ and simplify.

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Step 3.1.2.1

Divide each term in $7 A = 2$ by $7$ .

$\frac{7 A}{7} = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $7$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{7 A}{7} = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

Step 3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = 9 A + 7 B$

Step 3.2

Replace all occurrences of $A$ with $\frac{2}{7}$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $- 3 = 9 A + 7 B$ with $\frac{2}{7}$ .

$- 3 = 9 (\frac{2}{7}) + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $9 (\frac{2}{7})$ .

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Step 3.2.2.1.1

Combine $9$ and $\frac{2}{7}$ .

$- 3 = \frac{9 \cdot 2}{7} + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.2.2.1.2

Multiply $9$ by $2$ .

$- 3 = \frac{18}{7} + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = \frac{18}{7} + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = \frac{18}{7} + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

$- 3 = \frac{18}{7} + 7 B$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3

Solve for $B$ in $- 3 = \frac{18}{7} + 7 B$ .

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Step 3.3.1

Rewrite the equation as $\frac{18}{7} + 7 B = - 3$ .

$\frac{18}{7} + 7 B = - 3$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $\frac{18}{7}$ from both sides of the equation.

$7 B = - 3 - \frac{18}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$7 B = - 3 \cdot \frac{7}{7} - \frac{18}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.2.3

Combine $- 3$ and $\frac{7}{7}$ .

$7 B = \frac{- 3 \cdot 7}{7} - \frac{18}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.2.4

Combine the numerators over the common denominator.

$7 B = \frac{- 3 \cdot 7 - 18}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $- 3$ by $7$ .

$7 B = \frac{- 21 - 18}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.2.5.2

Subtract $18$ from $- 21$ .

$7 B = \frac{- 39}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

$7 B = \frac{- 39}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.2.6

Move the negative in front of the fraction.

$7 B = - \frac{39}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

$7 B = - \frac{39}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.3

Divide each term in $7 B = - \frac{39}{7}$ by $7$ and simplify.

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Step 3.3.3.1

Divide each term in $7 B = - \frac{39}{7}$ by $7$ .

$\frac{7 B}{7} = \frac{- \frac{39}{7}}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $7$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{7 B}{7} = \frac{- \frac{39}{7}}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- \frac{39}{7}}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

$B = \frac{- \frac{39}{7}}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

$B = \frac{- \frac{39}{7}}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$B = - \frac{39}{7} \cdot \frac{1}{7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.3.3.2

Multiply $- \frac{39}{7} \cdot \frac{1}{7}$ .

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Step 3.3.3.3.2.1

Multiply $\frac{1}{7}$ by $\frac{39}{7}$ .

$B = - \frac{39}{7 \cdot 7}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.3.3.3.2.2

Multiply $7$ by $7$ .

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$1 = 9 B + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$1 = 9 B + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$1 = 9 B + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$1 = 9 B + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$1 = 9 B + C$

Step 3.4

Replace all occurrences of $B$ with $- \frac{39}{49}$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $1 = 9 B + C$ with $- \frac{39}{49}$ .

$1 = 9 (- \frac{39}{49}) + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Multiply $9 (- \frac{39}{49})$ .

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Step 3.4.2.1.1.1

Multiply $- 1$ by $9$ .

$1 = - 9 (\frac{39}{49}) + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.4.2.1.1.2

Combine $- 9$ and $\frac{39}{49}$ .

$1 = \frac{- 9 \cdot 39}{49} + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.4.2.1.1.3

Multiply $- 9$ by $39$ .

$1 = \frac{- 351}{49} + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$1 = \frac{- 351}{49} + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.4.2.1.2

Move the negative in front of the fraction.

$1 = - \frac{351}{49} + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$1 = - \frac{351}{49} + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$1 = - \frac{351}{49} + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$1 = - \frac{351}{49} + C$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.5

Solve for $C$ in $1 = - \frac{351}{49} + C$ .

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Step 3.5.1

Rewrite the equation as $- \frac{351}{49} + C = 1$ .

$- \frac{351}{49} + C = 1$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.5.2

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

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Step 3.5.2.1

Add $\frac{351}{49}$ to both sides of the equation.

$C = 1 + \frac{351}{49}$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.5.2.2

Write $1$ as a fraction with a common denominator.

$C = \frac{49}{49} + \frac{351}{49}$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.5.2.3

Combine the numerators over the common denominator.

$C = \frac{49 + 351}{49}$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.5.2.4

Add $49$ and $351$ .

$C = \frac{400}{49}$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$C = \frac{400}{49}$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

$C = \frac{400}{49}$

$B = - \frac{39}{49}$

$A = \frac{2}{7}$

Step 3.6

Solve the system of equations.

$C = \frac{400}{49} B = - \frac{39}{49} A = \frac{2}{7}$

Step 3.7

List all of the solutions.

$C = \frac{400}{49}, B = - \frac{39}{49}, A = \frac{2}{7}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x^{2}} + \frac{B}{x} + \frac{C}{9 x + 7}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{\frac{2}{7}}{x^{2}} - \frac{\frac{39}{49}}{x} + \frac{\frac{400}{49}}{9 x + 7}$