Precalculus Examples

$\frac{2 x^{2} - 11 x + 5}{(x - 3) (x^{2} + 2 x - 5)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor by grouping.

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Step 1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 5 = 10$ and whose sum is $b = - 11$ .

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Step 1.1.1.1

Factor $- 11$ out of $- 11 x$ .

$\frac{2 x^{2} - 11 (x) + 5}{(x - 3) (x^{2} + 2 x - 5)}$

Step 1.1.1.2

Rewrite $- 11$ as $- 1$ plus $- 10$

$\frac{2 x^{2} + (- 1 - 10) x + 5}{(x - 3) (x^{2} + 2 x - 5)}$

Step 1.1.1.3

Apply the distributive property.

$\frac{2 x^{2} - 1 x - 10 x + 5}{(x - 3) (x^{2} + 2 x - 5)}$

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.

$\frac{(2 x^{2} - 1 x) - 10 x + 5}{(x - 3) (x^{2} + 2 x - 5)}$

Step 1.1.2.2

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

$\frac{x (2 x - 1) - 5 (2 x - 1)}{(x - 3) (x^{2} + 2 x - 5)}$

Step 1.1.3

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

$\frac{(2 x - 1) (x - 5)}{(x - 3) (x^{2} + 2 x - 5)}$

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 $A$ .

$\frac{A}{x - 3}$

Step 1.3

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 is 2nd order, $2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

$\frac{A}{x - 3} + \frac{B x + C}{x^{2} + 2 x - 5}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 3) (x^{2} + 2 x - 5)$ .

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

Step 1.5

Reduce the expression by cancelling the common factors.

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Step 1.5.1

Cancel the common factor of $x - 3$ .

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Step 1.5.1.1

Cancel the common factor.

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

Step 1.5.1.2

Rewrite the expression.

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

Step 1.5.2

Cancel the common factor of $x^{2} + 2 x - 5$ .

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Step 1.5.2.1

Cancel the common factor.

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

Step 1.5.2.2

Divide $(2 x - 1) (x - 5)$ by $1$ .

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

Step 1.6

Expand $(2 x - 1) (x - 5)$ using the FOIL Method.

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Step 1.6.1

Apply the distributive property.

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Apply the distributive property.

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

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

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

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Step 1.7.1.1.1

Move $x$ .

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

Step 1.7.1.1.2

Multiply $x$ by $x$ .

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

Step 1.7.1.2

Multiply $- 5$ by $2$ .

$2 x^{2} - 10 x - 1 x - 1 \cdot - 5 = \frac{(A) (x - 3) (x^{2} + 2 x - 5)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.7.1.3

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

$2 x^{2} - 10 x - x - 1 \cdot - 5 = \frac{(A) (x - 3) (x^{2} + 2 x - 5)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.7.1.4

Multiply $- 1$ by $- 5$ .

$2 x^{2} - 10 x - x + 5 = \frac{(A) (x - 3) (x^{2} + 2 x - 5)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.7.2

Subtract $x$ from $- 10 x$ .

$2 x^{2} - 11 x + 5 = \frac{(A) (x - 3) (x^{2} + 2 x - 5)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.8

Simplify each term.

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Step 1.8.1

Cancel the common factor of $x - 3$ .

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Step 1.8.1.1

Cancel the common factor.

$2 x^{2} - 11 x + 5 = \frac{A (x - 3) (x^{2} + 2 x - 5)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.8.1.2

Divide $(A) (x^{2} + 2 x - 5)$ by $1$ .

$2 x^{2} - 11 x + 5 = (A) (x^{2} + 2 x - 5) + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.8.2

Apply the distributive property.

$2 x^{2} - 11 x + 5 = A x^{2} + A (2 x) + A \cdot - 5 + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.8.3

Simplify.

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Step 1.8.3.1

Rewrite using the commutative property of multiplication.

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x + A \cdot - 5 + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.8.3.2

Move $- 5$ to the left of $A$ .

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 \cdot A + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.8.4

Cancel the common factor of $x^{2} + 2 x - 5$ .

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Step 1.8.4.1

Cancel the common factor.

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 A + \frac{(B x + C) (x - 3) (x^{2} + 2 x - 5)}{x^{2} + 2 x - 5}$

Step 1.8.4.2

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

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 A + (B x + C) (x - 3)$

Step 1.8.5

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

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Step 1.8.5.1

Apply the distributive property.

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 A + B x (x - 3) + C (x - 3)$

Step 1.8.5.2

Apply the distributive property.

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 A + B x \cdot x + B x \cdot - 3 + C (x - 3)$

Step 1.8.5.3

Apply the distributive property.

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 A + B x \cdot x + B x \cdot - 3 + C x + C \cdot - 3$

Step 1.8.6

Simplify each term.

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Step 1.8.6.1

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

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Step 1.8.6.1.1

Move $x$ .

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 A + B (x \cdot x) + B x \cdot - 3 + C x + C \cdot - 3$

Step 1.8.6.1.2

Multiply $x$ by $x$ .

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 A + B x^{2} + B x \cdot - 3 + C x + C \cdot - 3$

Step 1.8.6.2

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

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 A + B x^{2} - 3 \cdot (B x) + C x + C \cdot - 3$

Step 1.8.6.3

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

$2 x^{2} - 11 x + 5 = A x^{2} + 2 A x - 5 A + B x^{2} - 3 B x + C x - 3 C$

Step 1.9

Simplify the expression.

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Step 1.9.1

Move $A$ .

$2 x^{2} - 11 x + 5 = A x^{2} + 2 x A - 5 A + B x^{2} - 3 B x + C x - 3 C$

Step 1.9.2

Reorder $B$ and $x^{2}$ .

$2 x^{2} - 11 x + 5 = A x^{2} + 2 x A - 5 A + B x^{2} - 3 B x + C x - 3 C$

Step 1.9.3

Move $B$ .

$2 x^{2} - 11 x + 5 = A x^{2} + 2 x A - 5 A + B x^{2} - 3 x B + C x - 3 C$

Step 1.9.4

Move $- 5 A$ .

$2 x^{2} - 11 x + 5 = A x^{2} + 2 x A + B x^{2} - 3 x B + C x - 5 A - 3 C$

Step 1.9.5

Move $2 x A$ .

$2 x^{2} - 11 x + 5 = A x^{2} + B x^{2} + 2 x A - 3 x B + C x - 5 A - 3 C$

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.

$2 = A + B$

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.

$- 11 = 2 A - 3 B + C$

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.

$5 = - 5 A - 3 C$

Step 2.4

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

$2 = A + B$

$- 11 = 2 A - 3 B + C$

$5 = - 5 A - 3 C$

$2 = A + B$

$- 11 = 2 A - 3 B + C$

$5 = - 5 A - 3 C$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $2 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 2$ .

$A + B = 2$

$- 11 = 2 A - 3 B + C$

$5 = - 5 A - 3 C$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 2 - B$

$- 11 = 2 A - 3 B + C$

$5 = - 5 A - 3 C$

$A = 2 - B$

$- 11 = 2 A - 3 B + C$

$5 = - 5 A - 3 C$

Step 3.2

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

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Step 3.2.1

Replace all occurrences of $A$ in $- 11 = 2 A - 3 B + C$ with $2 - B$ .

$- 11 = 2 (2 - B) - 3 B + C$

$A = 2 - B$

$5 = - 5 A - 3 C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (2 - B) - 3 B + C$ .

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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.

$- 11 = 2 \cdot 2 + 2 (- B) - 3 B + C$

$A = 2 - B$

$5 = - 5 A - 3 C$

Step 3.2.2.1.1.2

Multiply $2$ by $2$ .

$- 11 = 4 + 2 (- B) - 3 B + C$

$A = 2 - B$

$5 = - 5 A - 3 C$

Step 3.2.2.1.1.3

Multiply $- 1$ by $2$ .

$- 11 = 4 - 2 B - 3 B + C$

$A = 2 - B$

$5 = - 5 A - 3 C$

$- 11 = 4 - 2 B - 3 B + C$

$A = 2 - B$

$5 = - 5 A - 3 C$

Step 3.2.2.1.2

Subtract $3 B$ from $- 2 B$ .

$- 11 = 4 - 5 B + C$

$A = 2 - B$

$5 = - 5 A - 3 C$

$- 11 = 4 - 5 B + C$

$A = 2 - B$

$5 = - 5 A - 3 C$

$- 11 = 4 - 5 B + C$

$A = 2 - B$

$5 = - 5 A - 3 C$

Step 3.2.3

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

$5 = - 5 (2 - B) - 3 C$

$- 11 = 4 - 5 B + C$

$A = 2 - B$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Apply the distributive property.

$5 = - 5 \cdot 2 - 5 (- B) - 3 C$

$- 11 = 4 - 5 B + C$

$A = 2 - B$

Step 3.2.4.1.2

Multiply $- 5$ by $2$ .

$5 = - 10 - 5 (- B) - 3 C$

$- 11 = 4 - 5 B + C$

$A = 2 - B$

Step 3.2.4.1.3

Multiply $- 1$ by $- 5$ .

$5 = - 10 + 5 B - 3 C$

$- 11 = 4 - 5 B + C$

$A = 2 - B$

$5 = - 10 + 5 B - 3 C$

$- 11 = 4 - 5 B + C$

$A = 2 - B$

$5 = - 10 + 5 B - 3 C$

$- 11 = 4 - 5 B + C$

$A = 2 - B$

$5 = - 10 + 5 B - 3 C$

$- 11 = 4 - 5 B + C$

$A = 2 - B$

Step 3.3

Reorder $2$ and $- B$ .

$A = - B + 2$

$5 = - 10 + 5 B - 3 C$

$- 11 = 4 - 5 B + C$

Step 3.4

Solve for $C$ in $- 11 = 4 - 5 B + C$ .

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Step 3.4.1

Rewrite the equation as $4 - 5 B + C = - 11$ .

$4 - 5 B + C = - 11$

$A = - B + 2$

$5 = - 10 + 5 B - 3 C$

Step 3.4.2

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

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Step 3.4.2.1

Subtract $4$ from both sides of the equation.

$- 5 B + C = - 11 - 4$

$A = - B + 2$

$5 = - 10 + 5 B - 3 C$

Step 3.4.2.2

Add $5 B$ to both sides of the equation.

$C = - 11 - 4 + 5 B$

$A = - B + 2$

$5 = - 10 + 5 B - 3 C$

Step 3.4.2.3

Subtract $4$ from $- 11$ .

$C = - 15 + 5 B$

$A = - B + 2$

$5 = - 10 + 5 B - 3 C$

$C = - 15 + 5 B$

$A = - B + 2$

$5 = - 10 + 5 B - 3 C$

$C = - 15 + 5 B$

$A = - B + 2$

$5 = - 10 + 5 B - 3 C$

Step 3.5

Replace all occurrences of $C$ with $- 15 + 5 B$ in each equation.

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Step 3.5.1

Replace all occurrences of $C$ in $5 = - 10 + 5 B - 3 C$ with $- 15 + 5 B$ .

$5 = - 10 + 5 B - 3 (- 15 + 5 B)$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $- 10 + 5 B - 3 (- 15 + 5 B)$ .

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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.

$5 = - 10 + 5 B - 3 \cdot - 15 - 3 (5 B)$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.5.2.1.1.2

Multiply $- 3$ by $- 15$ .

$5 = - 10 + 5 B + 45 - 3 (5 B)$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.5.2.1.1.3

Multiply $5$ by $- 3$ .

$5 = - 10 + 5 B + 45 - 15 B$

$C = - 15 + 5 B$

$A = - B + 2$

$5 = - 10 + 5 B + 45 - 15 B$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.5.2.1.2

Simplify by adding terms.

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Step 3.5.2.1.2.1

Add $- 10$ and $45$ .

$5 = 5 B + 35 - 15 B$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.5.2.1.2.2

Subtract $15 B$ from $5 B$ .

$5 = - 10 B + 35$

$C = - 15 + 5 B$

$A = - B + 2$

$5 = - 10 B + 35$

$C = - 15 + 5 B$

$A = - B + 2$

$5 = - 10 B + 35$

$C = - 15 + 5 B$

$A = - B + 2$

$5 = - 10 B + 35$

$C = - 15 + 5 B$

$A = - B + 2$

$5 = - 10 B + 35$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.6

Solve for $B$ in $5 = - 10 B + 35$ .

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Step 3.6.1

Rewrite the equation as $- 10 B + 35 = 5$ .

$- 10 B + 35 = 5$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.6.2

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

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Step 3.6.2.1

Subtract $35$ from both sides of the equation.

$- 10 B = 5 - 35$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.6.2.2

Subtract $35$ from $5$ .

$- 10 B = - 30$

$C = - 15 + 5 B$

$A = - B + 2$

$- 10 B = - 30$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.6.3

Divide each term in $- 10 B = - 30$ by $- 10$ and simplify.

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Step 3.6.3.1

Divide each term in $- 10 B = - 30$ by $- 10$ .

$\frac{- 10 B}{- 10} = \frac{- 30}{- 10}$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Cancel the common factor of $- 10$ .

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Step 3.6.3.2.1.1

Cancel the common factor.

$\frac{- 10 B}{- 10} = \frac{- 30}{- 10}$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.6.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 30}{- 10}$

$C = - 15 + 5 B$

$A = - B + 2$

$B = \frac{- 30}{- 10}$

$C = - 15 + 5 B$

$A = - B + 2$

$B = \frac{- 30}{- 10}$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Divide $- 30$ by $- 10$ .

$B = 3$

$C = - 15 + 5 B$

$A = - B + 2$

$B = 3$

$C = - 15 + 5 B$

$A = - B + 2$

$B = 3$

$C = - 15 + 5 B$

$A = - B + 2$

$B = 3$

$C = - 15 + 5 B$

$A = - B + 2$

Step 3.7

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

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Step 3.7.1

Replace all occurrences of $B$ in $C = - 15 + 5 B$ with $3$ .

$C = - 15 + 5 (3)$

$B = 3$

$A = - B + 2$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify $- 15 + 5 (3)$ .

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Step 3.7.2.1.1

Multiply $5$ by $3$ .

$C = - 15 + 15$

$B = 3$

$A = - B + 2$

Step 3.7.2.1.2

Add $- 15$ and $15$ .

$C = 0$

$B = 3$

$A = - B + 2$

$C = 0$

$B = 3$

$A = - B + 2$

$C = 0$

$B = 3$

$A = - B + 2$

Step 3.7.3

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

$A = - (3) + 2$

$C = 0$

$B = 3$

Step 3.7.4

Simplify the right side.

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Step 3.7.4.1

Simplify $- (3) + 2$ .

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Step 3.7.4.1.1

Multiply $- 1$ by $3$ .

$A = - 3 + 2$

$C = 0$

$B = 3$

Step 3.7.4.1.2

Add $- 3$ and $2$ .

$A = - 1$

$C = 0$

$B = 3$

$A = - 1$

$C = 0$

$B = 3$

$A = - 1$

$C = 0$

$B = 3$

$A = - 1$

$C = 0$

$B = 3$

Step 3.8

List all of the solutions.

$A = - 1, C = 0, B = 3$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x - 3} + \frac{B x + C}{x^{2} + 2 x - 5}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{- 1}{x - 3} + \frac{3 x + 0}{x^{2} + 2 x - 5}$

Step 5

Add $(3) \cdot x$ and $0$ .

$\frac{- 1}{x - 3} + \frac{3 x}{x^{2} + 2 x - 5}$