Precalculus Examples

$\frac{5 x^{2} - 18 x + 24}{x^{3} - 4 x^{2} + 4 x}$

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 $x$ out of $x^{3} - 4 x^{2} + 4 x$ .

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Step 1.1.1.1

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

$\frac{5 x^{2} - 18 x + 24}{x \cdot x^{2} - 4 x^{2} + 4 x}$

Step 1.1.1.2

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

$\frac{5 x^{2} - 18 x + 24}{x \cdot x^{2} + x (- 4 x) + 4 x}$

Step 1.1.1.3

Factor $x$ out of $4 x$ .

$\frac{5 x^{2} - 18 x + 24}{x \cdot x^{2} + x (- 4 x) + x \cdot 4}$

Step 1.1.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 4 x)$ .

$\frac{5 x^{2} - 18 x + 24}{x (x^{2} - 4 x) + x \cdot 4}$

Step 1.1.1.5

Factor $x$ out of $x (x^{2} - 4 x) + x \cdot 4$ .

$\frac{5 x^{2} - 18 x + 24}{x (x^{2} - 4 x + 4)}$

Step 1.1.2

Factor using the perfect square rule.

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Step 1.1.2.1

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

$\frac{5 x^{2} - 18 x + 24}{x (x^{2} - 4 x + 2^{2})}$

Step 1.1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 1.1.2.3

Rewrite the polynomial.

$\frac{5 x^{2} - 18 x + 24}{x (x^{2} - 2 \cdot x \cdot 2 + 2^{2})}$

Step 1.1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

$\frac{5 x^{2} - 18 x + 24}{x {(x - 2)}^{2}}$

Step 1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $B$ .

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

Step 1.3

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

Step 1.4

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

$\frac{(5 x^{2} - 18 x + 24) (x {(x - 2)}^{2})}{x {(x - 2)}^{2}} = \frac{A (x {(x - 2)}^{2})}{x} + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.5

Cancel the common factor of $x$ .

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Step 1.5.1

Cancel the common factor.

$\frac{(5 x^{2} - 18 x + 24) (x {(x - 2)}^{2})}{x {(x - 2)}^{2}} = \frac{A (x {(x - 2)}^{2})}{x} + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.5.2

Rewrite the expression.

$\frac{(5 x^{2} - 18 x + 24) ({(x - 2)}^{2})}{{(x - 2)}^{2}} = \frac{A (x {(x - 2)}^{2})}{x} + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.6

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

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Step 1.6.1

Cancel the common factor.

$\frac{(5 x^{2} - 18 x + 24) {(x - 2)}^{2}}{{(x - 2)}^{2}} = \frac{A (x {(x - 2)}^{2})}{x} + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.6.2

Divide $5 x^{2} - 18 x + 24$ by $1$ .

$5 x^{2} - 18 x + 24 = \frac{A (x {(x - 2)}^{2})}{x} + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of $x$ .

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Step 1.7.1.1

Cancel the common factor.

$5 x^{2} - 18 x + 24 = \frac{A (x {(x - 2)}^{2})}{x} + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.1.2

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

$5 x^{2} - 18 x + 24 = A ({(x - 2)}^{2}) + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.2

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

$5 x^{2} - 18 x + 24 = A ((x - 2) (x - 2)) + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.3

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

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Step 1.7.3.1

Apply the distributive property.

$5 x^{2} - 18 x + 24 = A (x (x - 2) - 2 (x - 2)) + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.3.2

Apply the distributive property.

$5 x^{2} - 18 x + 24 = A (x \cdot x + x \cdot - 2 - 2 (x - 2)) + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.3.3

Apply the distributive property.

$5 x^{2} - 18 x + 24 = A (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.4

Simplify and combine like terms.

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Step 1.7.4.1

Simplify each term.

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Step 1.7.4.1.1

Multiply $x$ by $x$ .

$5 x^{2} - 18 x + 24 = A (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.4.1.2

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

$5 x^{2} - 18 x + 24 = A (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.4.1.3

Multiply $- 2$ by $- 2$ .

$5 x^{2} - 18 x + 24 = A (x^{2} - 2 x - 2 x + 4) + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.4.2

Subtract $2 x$ from $- 2 x$ .

$5 x^{2} - 18 x + 24 = A (x^{2} - 4 x + 4) + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.5

Apply the distributive property.

$5 x^{2} - 18 x + 24 = A x^{2} + A (- 4 x) + A \cdot 4 + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.6

Simplify.

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Step 1.7.6.1

Rewrite using the commutative property of multiplication.

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + A \cdot 4 + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.6.2

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

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 \cdot A + \frac{(B) (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.7

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

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Step 1.7.7.1

Cancel the common factor.

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + \frac{B (x {(x - 2)}^{2})}{{(x - 2)}^{2}} + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.7.2

Divide $(B) (x)$ by $1$ .

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + (B) (x) + \frac{(C) (x {(x - 2)}^{2})}{x - 2}$

Step 1.7.8

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

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Step 1.7.8.1

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

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + \frac{(x - 2) (C (x (x - 2)))}{x - 2}$

Step 1.7.8.2

Cancel the common factors.

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Step 1.7.8.2.1

Multiply by $1$ .

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + \frac{(x - 2) (C (x (x - 2)))}{(x - 2) \cdot 1}$

Step 1.7.8.2.2

Cancel the common factor.

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + \frac{(x - 2) (C (x (x - 2)))}{(x - 2) \cdot 1}$

Step 1.7.8.2.3

Rewrite the expression.

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + \frac{C (x (x - 2))}{1}$

Step 1.7.8.2.4

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

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + C (x (x - 2))$

Step 1.7.9

Apply the distributive property.

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + C (x \cdot x + x \cdot - 2)$

Step 1.7.10

Multiply $x$ by $x$ .

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + C (x^{2} + x \cdot - 2)$

Step 1.7.11

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

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + C (x^{2} - 2 \cdot x)$

Step 1.7.12

Apply the distributive property.

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + C x^{2} + C (- 2 x)$

Step 1.7.13

Rewrite using the commutative property of multiplication.

$5 x^{2} - 18 x + 24 = A x^{2} - 4 A x + 4 A + B x + C x^{2} - 2 C x$

Step 1.8

Simplify the expression.

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Step 1.8.1

Move $A$ .

$5 x^{2} - 18 x + 24 = A x^{2} - 4 x A + 4 A + B x + C x^{2} - 2 C x$

Step 1.8.2

Reorder $B$ and $x$ .

$5 x^{2} - 18 x + 24 = A x^{2} - 4 x A + 4 A + B x + C x^{2} - 2 C x$

Step 1.8.3

Move $4 A$ .

$5 x^{2} - 18 x + 24 = A x^{2} - 4 x A + B x + C x^{2} - 2 C x + 4 A$

Step 1.8.4

Move $B x$ .

$5 x^{2} - 18 x + 24 = A x^{2} - 4 x A + C x^{2} + B x - 2 C x + 4 A$

Step 1.8.5

Move $- 4 x A$ .

$5 x^{2} - 18 x + 24 = A x^{2} + C x^{2} - 4 x A + B x - 2 C x + 4 A$

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.

$5 = A + C$

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.

$- 18 = - 4 A + B - 2 C$

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.

$24 = 4 A$

Step 2.4

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

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

$24 = 4 A$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

$24 = 4 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $24 = 4 A$ .

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Step 3.1.1

Rewrite the equation as $4 A = 24$ .

$4 A = 24$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

Step 3.1.2

Divide each term in $4 A = 24$ by $4$ and simplify.

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Step 3.1.2.1

Divide each term in $4 A = 24$ by $4$ .

$\frac{4 A}{4} = \frac{24}{4}$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $4$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{4 A}{4} = \frac{24}{4}$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

Step 3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{24}{4}$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

$A = \frac{24}{4}$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

$A = \frac{24}{4}$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $24$ by $4$ .

$A = 6$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

$A = 6$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

$A = 6$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

$A = 6$

$5 = A + C$

$- 18 = - 4 A + B - 2 C$

Step 3.2

Replace all occurrences of $A$ with $6$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $5 = A + C$ with $6$ .

$5 = (6) + C$

$A = 6$

$- 18 = - 4 A + B - 2 C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Remove parentheses.

$5 = 6 + C$

$A = 6$

$- 18 = - 4 A + B - 2 C$

$5 = 6 + C$

$A = 6$

$- 18 = - 4 A + B - 2 C$

Step 3.2.3

Replace all occurrences of $A$ in $- 18 = - 4 A + B - 2 C$ with $6$ .

$- 18 = - 4 \cdot 6 + B - 2 C$

$5 = 6 + C$

$A = 6$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Multiply $- 4$ by $6$ .

$- 18 = - 24 + B - 2 C$

$5 = 6 + C$

$A = 6$

$- 18 = - 24 + B - 2 C$

$5 = 6 + C$

$A = 6$

$- 18 = - 24 + B - 2 C$

$5 = 6 + C$

$A = 6$

Step 3.3

Solve for $C$ in $5 = 6 + C$ .

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Step 3.3.1

Rewrite the equation as $6 + C = 5$ .

$6 + C = 5$

$- 18 = - 24 + B - 2 C$

$A = 6$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $6$ from both sides of the equation.

$C = 5 - 6$

$- 18 = - 24 + B - 2 C$

$A = 6$

Step 3.3.2.2

Subtract $6$ from $5$ .

$C = - 1$

$- 18 = - 24 + B - 2 C$

$A = 6$

$C = - 1$

$- 18 = - 24 + B - 2 C$

$A = 6$

$C = - 1$

$- 18 = - 24 + B - 2 C$

$A = 6$

Step 3.4

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

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Step 3.4.1

Replace all occurrences of $C$ in $- 18 = - 24 + B - 2 C$ with $- 1$ .

$- 18 = - 24 + B - 2 \cdot - 1$

$C = - 1$

$A = 6$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 24 + B - 2 \cdot - 1$ .

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Step 3.4.2.1.1

Multiply $- 2$ by $- 1$ .

$- 18 = - 24 + B + 2$

$C = - 1$

$A = 6$

Step 3.4.2.1.2

Add $- 24$ and $2$ .

$- 18 = B - 22$

$C = - 1$

$A = 6$

$- 18 = B - 22$

$C = - 1$

$A = 6$

$- 18 = B - 22$

$C = - 1$

$A = 6$

$- 18 = B - 22$

$C = - 1$

$A = 6$

Step 3.5

Solve for $B$ in $- 18 = B - 22$ .

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Step 3.5.1

Rewrite the equation as $B - 22 = - 18$ .

$B - 22 = - 18$

$C = - 1$

$A = 6$

Step 3.5.2

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

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Step 3.5.2.1

Add $22$ to both sides of the equation.

$B = - 18 + 22$

$C = - 1$

$A = 6$

Step 3.5.2.2

Add $- 18$ and $22$ .

$B = 4$

$C = - 1$

$A = 6$

$B = 4$

$C = - 1$

$A = 6$

$B = 4$

$C = - 1$

$A = 6$

Step 3.6

Solve the system of equations.

$B = 4 C = - 1 A = 6$

Step 3.7

List all of the solutions.

$B = 4, C = - 1, A = 6$

Step 4

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

$\frac{6}{x} + \frac{4}{{(x - 2)}^{2}} + \frac{- 1}{x - 2}$