Precalculus Examples

$\frac{80 x^{3} + 24 x^{2} - 2 x + 11}{40 x^{2} + 12 x - 16}$

Step 1

Divide using long polynomial division.

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Step 1.1

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

$40 x^{2}$

$12 x$

$16$

$80 x^{3}$

$24 x^{2}$

$2 x$

$11$

Step 1.2

Divide the highest order term in the dividend $80 x^{3}$ by the highest order term in divisor $40 x^{2}$ .

$2 x$

$40 x^{2}$

$12 x$

$16$

$80 x^{3}$

$24 x^{2}$

$2 x$

$11$

Step 1.3

Multiply the new quotient term by the divisor.

$2 x$

$40 x^{2}$

$12 x$

$16$

$80 x^{3}$

$24 x^{2}$

$2 x$

$11$

$80 x^{3}$

$24 x^{2}$

$32 x$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $80 x^{3} + 24 x^{2} - 32 x$

$2 x$

$40 x^{2}$

$12 x$

$16$

$80 x^{3}$

$24 x^{2}$

$2 x$

$11$

$80 x^{3}$

$24 x^{2}$

$32 x$

Step 1.5

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

$2 x$

$40 x^{2}$

$12 x$

$16$

$80 x^{3}$

$24 x^{2}$

$2 x$

$11$

$80 x^{3}$

$24 x^{2}$

$32 x$

$30 x$

Step 1.6

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

$2 x$

$40 x^{2}$

$12 x$

$16$

$80 x^{3}$

$24 x^{2}$

$2 x$

$11$

$80 x^{3}$

$24 x^{2}$

$32 x$

$30 x$

$11$

Step 1.7

The final answer is the quotient plus the remainder over the divisor.

$2 x + \frac{30 x + 11}{40 x^{2} + 12 x - 16}$

Step 2

Decompose the fraction and multiply through by the common denominator.

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Step 2.1

Factor the fraction.

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Step 2.1.1

Factor $4$ out of $40 x^{2} + 12 x - 16$ .

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Step 2.1.1.1

Factor $4$ out of $40 x^{2}$ .

$\frac{30 x + 11}{4 (10 x^{2}) + 12 x - 16}$

Step 2.1.1.2

Factor $4$ out of $12 x$ .

$\frac{30 x + 11}{4 (10 x^{2}) + 4 (3 x) - 16}$

Step 2.1.1.3

Factor $4$ out of $- 16$ .

$\frac{30 x + 11}{4 (10 x^{2}) + 4 (3 x) + 4 (- 4)}$

Step 2.1.1.4

Factor $4$ out of $4 (10 x^{2}) + 4 (3 x)$ .

$\frac{30 x + 11}{4 (10 x^{2} + 3 x) + 4 (- 4)}$

Step 2.1.1.5

Factor $4$ out of $4 (10 x^{2} + 3 x) + 4 (- 4)$ .

$\frac{30 x + 11}{4 (10 x^{2} + 3 x - 4)}$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor by grouping.

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Step 2.1.2.1.1

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 $a \cdot c = 10 \cdot - 4 = - 40$ and whose sum is $b = 3$ .

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Step 2.1.2.1.1.1

Factor $3$ out of $3 x$ .

$\frac{30 x + 11}{4 (10 x^{2} + 3 (x) - 4)}$

Step 2.1.2.1.1.2

Rewrite $3$ as $- 5$ plus $8$

$\frac{30 x + 11}{4 (10 x^{2} + (- 5 + 8) x - 4)}$

Step 2.1.2.1.1.3

Apply the distributive property.

$\frac{30 x + 11}{4 (10 x^{2} - 5 x + 8 x - 4)}$

Step 2.1.2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1.2.1

Group the first two terms and the last two terms.

$\frac{30 x + 11}{4 ((10 x^{2} - 5 x) + 8 x - 4)}$

Step 2.1.2.1.2.2

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

$\frac{30 x + 11}{4 (5 x (2 x - 1) + 4 (2 x - 1))}$

Step 2.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{30 x + 11}{4 ((2 x - 1) (5 x + 4))}$

Step 2.1.2.2

Remove unnecessary parentheses.

$\frac{30 x + 11}{4 (2 x - 1) (5 x + 4)}$

Step 2.2

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 $A$ .

$\frac{A}{2 x - 1}$

Step 2.3

$\frac{A}{2 x - 1} + \frac{B}{5 x + 4}$

Step 2.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $4 (2 x - 1) (5 x + 4)$ .

$\frac{(30 x + 11) (4 (2 x - 1) (5 x + 4))}{4 (2 x - 1) (5 x + 4)} = \frac{(A) (4 (2 x - 1) (5 x + 4))}{2 x - 1} + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.5

Cancel the common factor of $4$ .

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Step 2.5.1

Cancel the common factor.

$\frac{(30 x + 11) (4 (2 x - 1) (5 x + 4))}{4 (2 x - 1) (5 x + 4)} = \frac{(A) (4 (2 x - 1) (5 x + 4))}{2 x - 1} + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.5.2

Rewrite the expression.

$\frac{(30 x + 11) ((2 x - 1) (5 x + 4))}{(2 x - 1) (5 x + 4)} = \frac{(A) (4 (2 x - 1) (5 x + 4))}{2 x - 1} + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.6

Cancel the common factor of $2 x - 1$ .

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Step 2.6.1

Cancel the common factor.

$\frac{(30 x + 11) ((2 x - 1) (5 x + 4))}{(2 x - 1) (5 x + 4)} = \frac{(A) (4 (2 x - 1) (5 x + 4))}{2 x - 1} + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.6.2

Rewrite the expression.

$\frac{(30 x + 11) (5 x + 4)}{5 x + 4} = \frac{(A) (4 (2 x - 1) (5 x + 4))}{2 x - 1} + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.7

Cancel the common factor of $5 x + 4$ .

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Step 2.7.1

Cancel the common factor.

$\frac{(30 x + 11) (5 x + 4)}{5 x + 4} = \frac{(A) (4 (2 x - 1) (5 x + 4))}{2 x - 1} + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.7.2

Divide $30 x + 11$ by $1$ .

$30 x + 11 = \frac{(A) (4 (2 x - 1) (5 x + 4))}{2 x - 1} + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.8

Simplify each term.

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Step 2.8.1

Cancel the common factor of $2 x - 1$ .

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Step 2.8.1.1

Cancel the common factor.

$30 x + 11 = \frac{A (4 (2 x - 1) (5 x + 4))}{2 x - 1} + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.8.1.2

Divide $(A) (4 (5 x + 4))$ by $1$ .

$30 x + 11 = (A) (4 (5 x + 4)) + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.8.2

Rewrite using the commutative property of multiplication.

$30 x + 11 = 4 A (5 x + 4) + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.8.3

Apply the distributive property.

$30 x + 11 = 4 A (5 x) + 4 A \cdot 4 + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.8.4

Rewrite using the commutative property of multiplication.

$30 x + 11 = 4 \cdot (5 A x) + 4 A \cdot 4 + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.8.5

Multiply $4$ by $4$ .

$30 x + 11 = 4 \cdot (5 A x) + 16 A + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.8.6

Multiply $4$ by $5$ .

$30 x + 11 = 20 A x + 16 A + \frac{(B) (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.8.7

Cancel the common factor of $5 x + 4$ .

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Step 2.8.7.1

Cancel the common factor.

$30 x + 11 = 20 A x + 16 A + \frac{B (4 (2 x - 1) (5 x + 4))}{5 x + 4}$

Step 2.8.7.2

Divide $(B) (4 (2 x - 1))$ by $1$ .

$30 x + 11 = 20 A x + 16 A + (B) (4 (2 x - 1))$

Step 2.8.8

Rewrite using the commutative property of multiplication.

$30 x + 11 = 20 A x + 16 A + 4 B (2 x - 1)$

Step 2.8.9

Apply the distributive property.

$30 x + 11 = 20 A x + 16 A + 4 B (2 x) + 4 B \cdot - 1$

Step 2.8.10

Rewrite using the commutative property of multiplication.

$30 x + 11 = 20 A x + 16 A + 4 \cdot (2 B x) + 4 B \cdot - 1$

Step 2.8.11

Multiply $- 1$ by $4$ .

$30 x + 11 = 20 A x + 16 A + 4 \cdot (2 B x) - 4 B$

Step 2.8.12

Multiply $4$ by $2$ .

$30 x + 11 = 20 A x + 16 A + 8 B x - 4 B$

Step 2.9

Reorder.

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Step 2.9.1

Move $A$ .

$30 x + 11 = 20 x A + 16 A + 8 B x - 4 B$

Step 2.9.2

Move $B$ .

$30 x + 11 = 20 x A + 16 A + 8 x B - 4 B$

Step 2.9.3

Move $16 A$ .

$30 x + 11 = 20 x A + 8 x B + 16 A - 4 B$

Step 3

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

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Step 3.1

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.

$30 = 20 A + 8 B$

Step 3.2

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.

$11 = 16 A - 4 B$

Step 3.3

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

$30 = 20 A + 8 B$

$11 = 16 A - 4 B$

$30 = 20 A + 8 B$

$11 = 16 A - 4 B$

Step 4

Solve the system of equations.

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Step 4.1

Solve for $A$ in $30 = 20 A + 8 B$ .

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Step 4.1.1

Rewrite the equation as $20 A + 8 B = 30$ .

$20 A + 8 B = 30$

$11 = 16 A - 4 B$

Step 4.1.2

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

$20 A = 30 - 8 B$

$11 = 16 A - 4 B$

Step 4.1.3

Divide each term in $20 A = 30 - 8 B$ by $20$ and simplify.

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Step 4.1.3.1

Divide each term in $20 A = 30 - 8 B$ by $20$ .

$\frac{20 A}{20} = \frac{30}{20} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

Step 4.1.3.2

Simplify the left side.

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Step 4.1.3.2.1

Cancel the common factor of $20$ .

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Step 4.1.3.2.1.1

Cancel the common factor.

$\frac{20 A}{20} = \frac{30}{20} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

Step 4.1.3.2.1.2

Divide $A$ by $1$ .

$A = \frac{30}{20} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

$A = \frac{30}{20} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

$A = \frac{30}{20} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

Step 4.1.3.3

Simplify the right side.

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Step 4.1.3.3.1

Simplify each term.

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Step 4.1.3.3.1.1

Cancel the common factor of $30$ and $20$ .

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Step 4.1.3.3.1.1.1

Factor $10$ out of $30$ .

$A = \frac{10 (3)}{20} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

Step 4.1.3.3.1.1.2

Cancel the common factors.

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Step 4.1.3.3.1.1.2.1

Factor $10$ out of $20$ .

$A = \frac{10 \cdot 3}{10 \cdot 2} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

Step 4.1.3.3.1.1.2.2

Cancel the common factor.

$A = \frac{10 \cdot 3}{10 \cdot 2} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

Step 4.1.3.3.1.1.2.3

Rewrite the expression.

$A = \frac{3}{2} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

$A = \frac{3}{2} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

$A = \frac{3}{2} + \frac{- 8 B}{20}$

$11 = 16 A - 4 B$

Step 4.1.3.3.1.2

Cancel the common factor of $- 8$ and $20$ .

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Step 4.1.3.3.1.2.1

Factor $4$ out of $- 8 B$ .

$A = \frac{3}{2} + \frac{4 (- 2 B)}{20}$

$11 = 16 A - 4 B$

Step 4.1.3.3.1.2.2

Cancel the common factors.

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Step 4.1.3.3.1.2.2.1

Factor $4$ out of $20$ .

$A = \frac{3}{2} + \frac{4 (- 2 B)}{4 (5)}$

$11 = 16 A - 4 B$

Step 4.1.3.3.1.2.2.2

Cancel the common factor.

$A = \frac{3}{2} + \frac{4 (- 2 B)}{4 \cdot 5}$

$11 = 16 A - 4 B$

Step 4.1.3.3.1.2.2.3

Rewrite the expression.

$A = \frac{3}{2} + \frac{- 2 B}{5}$

$11 = 16 A - 4 B$

$A = \frac{3}{2} + \frac{- 2 B}{5}$

$11 = 16 A - 4 B$

$A = \frac{3}{2} + \frac{- 2 B}{5}$

$11 = 16 A - 4 B$

Step 4.1.3.3.1.3

Move the negative in front of the fraction.

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 16 A - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 16 A - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 16 A - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 16 A - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 16 A - 4 B$

Step 4.2

Replace all occurrences of $A$ with $\frac{3}{2} - \frac{2 B}{5}$ in each equation.

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Step 4.2.1

Replace all occurrences of $A$ in $11 = 16 A - 4 B$ with $\frac{3}{2} - \frac{2 B}{5}$ .

$11 = 16 (\frac{3}{2} - \frac{2 B}{5}) - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $16 (\frac{3}{2} - \frac{2 B}{5}) - 4 B$ .

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Step 4.2.2.1.1

Simplify each term.

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Step 4.2.2.1.1.1

Apply the distributive property.

$11 = 16 (\frac{3}{2}) + 16 (- \frac{2 B}{5}) - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Factor $2$ out of $16$ .

$11 = 2 (8) (\frac{3}{2}) + 16 (- \frac{2 B}{5}) - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.1.2.2

Cancel the common factor.

$11 = 2 \cdot (8 (\frac{3}{2})) + 16 (- \frac{2 B}{5}) - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.1.2.3

Rewrite the expression.

$11 = 8 \cdot 3 + 16 (- \frac{2 B}{5}) - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 8 \cdot 3 + 16 (- \frac{2 B}{5}) - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.1.3

Multiply $8$ by $3$ .

$11 = 24 + 16 (- \frac{2 B}{5}) - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.1.4

Multiply $16 (- \frac{2 B}{5})$ .

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Step 4.2.2.1.1.4.1

Multiply $- 1$ by $16$ .

$11 = 24 - 16 \frac{2 B}{5} - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.1.4.2

Combine $- 16$ and $\frac{2 B}{5}$ .

$11 = 24 + \frac{- 16 (2 B)}{5} - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.1.4.3

Multiply $2$ by $- 16$ .

$11 = 24 + \frac{- 32 B}{5} - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 24 + \frac{- 32 B}{5} - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.1.5

Move the negative in front of the fraction.

$11 = 24 - \frac{32 B}{5} - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 24 - \frac{32 B}{5} - 4 B$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.2

To write $- 4 B$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$11 = 24 - \frac{32 B}{5} - 4 B \cdot \frac{5}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.3

Simplify terms.

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Step 4.2.2.1.3.1

Combine $- 4 B$ and $\frac{5}{5}$ .

$11 = 24 - \frac{32 B}{5} + \frac{- 4 B \cdot 5}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.3.2

Combine the numerators over the common denominator.

$11 = 24 + \frac{- 32 B - 4 B \cdot 5}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 24 + \frac{- 32 B - 4 B \cdot 5}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.4

Simplify each term.

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Step 4.2.2.1.4.1

Simplify the numerator.

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Step 4.2.2.1.4.1.1

Factor $4 B$ out of $- 32 B - 4 B \cdot 5$ .

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Step 4.2.2.1.4.1.1.1

Factor $4 B$ out of $- 32 B$ .

$11 = 24 + \frac{4 B (- 8) - 4 B \cdot 5}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.4.1.1.2

Factor $4 B$ out of $- 4 B \cdot 5$ .

$11 = 24 + \frac{4 B (- 8) + 4 B (- 1 \cdot 5)}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.4.1.1.3

Factor $4 B$ out of $4 B (- 8) + 4 B (- 1 \cdot 5)$ .

$11 = 24 + \frac{4 B (- 8 - 1 \cdot 5)}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 24 + \frac{4 B (- 8 - 1 \cdot 5)}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.4.1.2

Multiply $- 1$ by $5$ .

$11 = 24 + \frac{4 B (- 8 - 5)}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.4.1.3

Subtract $5$ from $- 8$ .

$11 = 24 + \frac{4 B \cdot - 13}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.4.1.4

Multiply $- 13$ by $4$ .

$11 = 24 + \frac{- 52 B}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 24 + \frac{- 52 B}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.2.2.1.4.2

Move the negative in front of the fraction.

$11 = 24 - \frac{52 B}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 24 - \frac{52 B}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 24 - \frac{52 B}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 24 - \frac{52 B}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$11 = 24 - \frac{52 B}{5}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3

Solve for $B$ in $11 = 24 - \frac{52 B}{5}$ .

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Step 4.3.1

Rewrite the equation as $24 - \frac{52 B}{5} = 11$ .

$24 - \frac{52 B}{5} = 11$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.2

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

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Step 4.3.2.1

Subtract $24$ from both sides of the equation.

$- \frac{52 B}{5} = 11 - 24$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.2.2

Subtract $24$ from $11$ .

$- \frac{52 B}{5} = - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

$- \frac{52 B}{5} = - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.3

Multiply both sides of the equation by $- \frac{5}{52}$ .

$- \frac{5}{52} \cdot (- \frac{52 B}{5}) = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4

Simplify both sides of the equation.

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Step 4.3.4.1

Simplify the left side.

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Step 4.3.4.1.1

Simplify $- \frac{5}{52} (- \frac{52 B}{5})$ .

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Step 4.3.4.1.1.1

Cancel the common factor of $5$ .

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Step 4.3.4.1.1.1.1

Move the leading negative in $- \frac{5}{52}$ into the numerator.

$\frac{- 5}{52} \cdot (- \frac{52 B}{5}) = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.1.1.1.2

Move the leading negative in $- \frac{52 B}{5}$ into the numerator.

$\frac{- 5}{52} \cdot \frac{- 52 B}{5} = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{52} \cdot \frac{- 52 B}{5} = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{52} \cdot \frac{- 52 B}{5} = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{52} \cdot (- 52 B) = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

$\frac{- 1}{52} \cdot (- 52 B) = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.1.1.2

Cancel the common factor of $52$ .

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Step 4.3.4.1.1.2.1

Factor $52$ out of $- 52 B$ .

$\frac{- 1}{52} \cdot (52 (- B)) = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{52} \cdot (52 (- B)) = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.1.1.2.3

Rewrite the expression.

$B = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

$B = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.1.1.3

Multiply.

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Step 4.3.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 B = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.1.1.3.2

Multiply $B$ by $1$ .

$B = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

$B = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

$B = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

$B = - \frac{5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.2

Simplify the right side.

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Step 4.3.4.2.1

Simplify $- \frac{5}{52} \cdot - 13$ .

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Step 4.3.4.2.1.1

Cancel the common factor of $13$ .

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Step 4.3.4.2.1.1.1

Move the leading negative in $- \frac{5}{52}$ into the numerator.

$B = \frac{- 5}{52} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.2.1.1.2

Factor $13$ out of $52$ .

$B = \frac{- 5}{13 (4)} \cdot - 13$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.2.1.1.3

Factor $13$ out of $- 13$ .

$B = \frac{- 5}{13 \cdot 4} \cdot (13 \cdot - 1)$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.2.1.1.4

Cancel the common factor.

$B = \frac{- 5}{13 \cdot 4} \cdot (13 \cdot - 1)$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.2.1.1.5

Rewrite the expression.

$B = \frac{- 5}{4} \cdot - 1$

$A = \frac{3}{2} - \frac{2 B}{5}$

$B = \frac{- 5}{4} \cdot - 1$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.2.1.2

Combine $\frac{- 5}{4}$ and $- 1$ .

$B = \frac{- 5 \cdot - 1}{4}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.3.4.2.1.3

Multiply $- 5$ by $- 1$ .

$B = \frac{5}{4}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$B = \frac{5}{4}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$B = \frac{5}{4}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$B = \frac{5}{4}$

$A = \frac{3}{2} - \frac{2 B}{5}$

$B = \frac{5}{4}$

$A = \frac{3}{2} - \frac{2 B}{5}$

Step 4.4

Replace all occurrences of $B$ with $\frac{5}{4}$ in each equation.

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Step 4.4.1

Replace all occurrences of $B$ in $A = \frac{3}{2} - \frac{2 B}{5}$ with $\frac{5}{4}$ .

$A = \frac{3}{2} - \frac{2 (\frac{5}{4})}{5}$

$B = \frac{5}{4}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $\frac{3}{2} - \frac{2 (\frac{5}{4})}{5}$ .

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Step 4.4.2.1.1

Simplify each term.

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Step 4.4.2.1.1.1

Combine $2$ and $\frac{5}{4}$ .

$A = \frac{3}{2} - \frac{\frac{2 \cdot 5}{4}}{5}$

$B = \frac{5}{4}$

Step 4.4.2.1.1.2

Multiply $2$ by $5$ .

$A = \frac{3}{2} - \frac{\frac{10}{4}}{5}$

$B = \frac{5}{4}$

Step 4.4.2.1.1.3

Cancel the common factor of $10$ and $4$ .

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Step 4.4.2.1.1.3.1

Factor $2$ out of $10$ .

$A = \frac{3}{2} - \frac{\frac{2 (5)}{4}}{5}$

$B = \frac{5}{4}$

Step 4.4.2.1.1.3.2

Cancel the common factors.

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Step 4.4.2.1.1.3.2.1

Factor $2$ out of $4$ .

$A = \frac{3}{2} - \frac{\frac{2 \cdot 5}{2 \cdot 2}}{5}$

$B = \frac{5}{4}$

Step 4.4.2.1.1.3.2.2

Cancel the common factor.

$A = \frac{3}{2} - \frac{\frac{2 \cdot 5}{2 \cdot 2}}{5}$

$B = \frac{5}{4}$

Step 4.4.2.1.1.3.2.3

Rewrite the expression.

$A = \frac{3}{2} - \frac{\frac{5}{2}}{5}$

$B = \frac{5}{4}$

$A = \frac{3}{2} - \frac{\frac{5}{2}}{5}$

$B = \frac{5}{4}$

$A = \frac{3}{2} - \frac{\frac{5}{2}}{5}$

$B = \frac{5}{4}$

Step 4.4.2.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$A = \frac{3}{2} - (\frac{5}{2} \cdot \frac{1}{5})$

$B = \frac{5}{4}$

Step 4.4.2.1.1.5

Cancel the common factor of $5$ .

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Step 4.4.2.1.1.5.1

Cancel the common factor.

$A = \frac{3}{2} - (\frac{5}{2} \cdot \frac{1}{5})$

$B = \frac{5}{4}$

Step 4.4.2.1.1.5.2

Rewrite the expression.

$A = \frac{3}{2} - \frac{1}{2}$

$B = \frac{5}{4}$

$A = \frac{3}{2} - \frac{1}{2}$

$B = \frac{5}{4}$

$A = \frac{3}{2} - \frac{1}{2}$

$B = \frac{5}{4}$

Step 4.4.2.1.2

Combine fractions.

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Step 4.4.2.1.2.1

Combine the numerators over the common denominator.

$A = \frac{3 - 1}{2}$

$B = \frac{5}{4}$

Step 4.4.2.1.2.2

Simplify the expression.

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Step 4.4.2.1.2.2.1

Subtract $1$ from $3$ .

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

$B = \frac{5}{4}$

Step 4.4.2.1.2.2.2

Divide $2$ by $2$ .

$A = 1$

$B = \frac{5}{4}$

$A = 1$

$B = \frac{5}{4}$

$A = 1$

$B = \frac{5}{4}$

$A = 1$

$B = \frac{5}{4}$

$A = 1$

$B = \frac{5}{4}$

$A = 1$

$B = \frac{5}{4}$

Step 4.5

List all of the solutions.

$A = 1, B = \frac{5}{4}$

Step 5

Replace each of the partial fraction coefficients in $2 x + \frac{A}{2 x - 1} + \frac{B}{5 x + 4}$ with the values found for $A$ and $B$ .

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