Precalculus Examples

$\frac{x^{3} - 4 x^{2} + 9 x - 5}{{(x^{2} - 2 x + 3)}^{2}}$

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 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.

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

Step 1.2

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

Step 1.3

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

$\frac{(x^{3} - 4 x^{2} + 9 x - 5) {(x^{2} - 2 x + 3)}^{2}}{{(x^{2} - 2 x + 3)}^{2}} = \frac{(A x + B) {(x^{2} - 2 x + 3)}^{2}}{{(x^{2} - 2 x + 3)}^{2}} + \frac{(C x + D) {(x^{2} - 2 x + 3)}^{2}}{x^{2} - 2 x + 3}$

Step 1.4

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

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Step 1.4.1

Cancel the common factor.

Step 1.4.2

Divide $x^{3} - 4 x^{2} + 9 x - 5$ by $1$ .

$x^{3} - 4 x^{2} + 9 x - 5 = \frac{(A x + B) {(x^{2} - 2 x + 3)}^{2}}{{(x^{2} - 2 x + 3)}^{2}} + \frac{(C x + D) {(x^{2} - 2 x + 3)}^{2}}{x^{2} - 2 x + 3}$

Step 1.5

Simplify each term.

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Step 1.5.1

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

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Step 1.5.1.1

Cancel the common factor.

$x^{3} - 4 x^{2} + 9 x - 5 = \frac{(A x + B) {(x^{2} - 2 x + 3)}^{2}}{{(x^{2} - 2 x + 3)}^{2}} + \frac{(C x + D) {(x^{2} - 2 x + 3)}^{2}}{x^{2} - 2 x + 3}$

Step 1.5.1.2

Divide $A x + B$ by $1$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + \frac{(C x + D) {(x^{2} - 2 x + 3)}^{2}}{x^{2} - 2 x + 3}$

Step 1.5.2

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

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Step 1.5.2.1

Factor $x^{2} - 2 x + 3$ out of $(C x + D) {(x^{2} - 2 x + 3)}^{2}$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + \frac{(x^{2} - 2 x + 3) ((C x + D) (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Multiply by $1$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + \frac{(x^{2} - 2 x + 3) ((C x + D) (x^{2} - 2 x + 3))}{(x^{2} - 2 x + 3) \cdot 1}$

Step 1.5.2.2.2

Cancel the common factor.

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + \frac{(x^{2} - 2 x + 3) ((C x + D) (x^{2} - 2 x + 3))}{(x^{2} - 2 x + 3) \cdot 1}$

Step 1.5.2.2.3

Rewrite the expression.

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + \frac{(C x + D) (x^{2} - 2 x + 3)}{1}$

Step 1.5.2.2.4

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

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + (C x + D) (x^{2} - 2 x + 3)$

Step 1.5.3

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

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C x \cdot x^{2} + C x (- 2 x) + C x \cdot 3 + D x^{2} + D (- 2 x) + D \cdot 3$

Step 1.5.4

Simplify each term.

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Step 1.5.4.1

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

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Step 1.5.4.1.1

Move $x^{2}$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C (x^{2} x) + C x (- 2 x) + C x \cdot 3 + D x^{2} + D (- 2 x) + D \cdot 3$

Step 1.5.4.1.2

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

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Step 1.5.4.1.2.1

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

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C (x^{2} x) + C x (- 2 x) + C x \cdot 3 + D x^{2} + D (- 2 x) + D \cdot 3$

Step 1.5.4.1.2.2

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

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C x^{2 + 1} + C x (- 2 x) + C x \cdot 3 + D x^{2} + D (- 2 x) + D \cdot 3$

Step 1.5.4.1.3

Add $2$ and $1$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C x^{3} + C x (- 2 x) + C x \cdot 3 + D x^{2} + D (- 2 x) + D \cdot 3$

Step 1.5.4.2

Rewrite using the commutative property of multiplication.

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C x^{3} - 2 C x \cdot x + C x \cdot 3 + D x^{2} + D (- 2 x) + D \cdot 3$

Step 1.5.4.3

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

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Step 1.5.4.3.1

Move $x$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C x^{3} - 2 C (x \cdot x) + C x \cdot 3 + D x^{2} + D (- 2 x) + D \cdot 3$

Step 1.5.4.3.2

Multiply $x$ by $x$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C x^{3} - 2 C x^{2} + C x \cdot 3 + D x^{2} + D (- 2 x) + D \cdot 3$

Step 1.5.4.4

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

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C x^{3} - 2 C x^{2} + 3 \cdot (C x) + D x^{2} + D (- 2 x) + D \cdot 3$

Step 1.5.4.5

Rewrite using the commutative property of multiplication.

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C x^{3} - 2 C x^{2} + 3 C x + D x^{2} - 2 D x + D \cdot 3$

Step 1.5.4.6

Move $3$ to the left of $D$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + B + C x^{3} - 2 C x^{2} + 3 C x + D x^{2} - 2 D x + 3 D$

Step 1.6

Simplify the expression.

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Step 1.6.1

Move $B$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + C x^{3} - 2 C x^{2} + 3 C x + D x^{2} - 2 D x + B + 3 D$

Step 1.6.2

Move $3 C x$ .

$x^{3} - 4 x^{2} + 9 x - 5 = A x + C x^{3} - 2 C x^{2} + D x^{2} + 3 C x - 2 D x + B + 3 D$

Step 1.6.3

Move $A x$ .

$x^{3} - 4 x^{2} + 9 x - 5 = C x^{3} - 2 C x^{2} + D x^{2} + A x + 3 C x - 2 D x + B + 3 D$

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.

$1 = C$

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.

$- 4 = - 2 C + D$

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.

$9 = A + 3 C - 2 D$

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.

$- 5 = B + 3 D$

Step 2.5

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

$1 = C$

$- 4 = - 2 C + D$

$9 = A + 3 C - 2 D$

$- 5 = B + 3 D$

$1 = C$

$- 4 = - 2 C + D$

$9 = A + 3 C - 2 D$

$- 5 = B + 3 D$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $C = 1$ .

$C = 1$

$- 4 = - 2 C + D$

$9 = A + 3 C - 2 D$

$- 5 = B + 3 D$

Step 3.2

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

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Step 3.2.1

Replace all occurrences of $C$ in $- 4 = - 2 C + D$ with $1$ .

$- 4 = - 2 \cdot 1 + D$

$C = 1$

$9 = A + 3 C - 2 D$

$- 5 = B + 3 D$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 2$ by $1$ .

$- 4 = - 2 + D$

$C = 1$

$9 = A + 3 C - 2 D$

$- 5 = B + 3 D$

$- 4 = - 2 + D$

$C = 1$

$9 = A + 3 C - 2 D$

$- 5 = B + 3 D$

Step 3.2.3

Replace all occurrences of $C$ in $9 = A + 3 C - 2 D$ with $1$ .

$9 = A + 3 (1) - 2 D$

$- 4 = - 2 + D$

$C = 1$

$- 5 = B + 3 D$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Multiply $3$ by $1$ .

$9 = A + 3 - 2 D$

$- 4 = - 2 + D$

$C = 1$

$- 5 = B + 3 D$

$9 = A + 3 - 2 D$

$- 4 = - 2 + D$

$C = 1$

$- 5 = B + 3 D$

$9 = A + 3 - 2 D$

$- 4 = - 2 + D$

$C = 1$

$- 5 = B + 3 D$

Step 3.3

Solve for $D$ in $- 4 = - 2 + D$ .

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Step 3.3.1

Rewrite the equation as $- 2 + D = - 4$ .

$- 2 + D = - 4$

$9 = A + 3 - 2 D$

$C = 1$

$- 5 = B + 3 D$

Step 3.3.2

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

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Step 3.3.2.1

Add $2$ to both sides of the equation.

$D = - 4 + 2$

$9 = A + 3 - 2 D$

$C = 1$

$- 5 = B + 3 D$

Step 3.3.2.2

Add $- 4$ and $2$ .

$D = - 2$

$9 = A + 3 - 2 D$

$C = 1$

$- 5 = B + 3 D$

$D = - 2$

$9 = A + 3 - 2 D$

$C = 1$

$- 5 = B + 3 D$

$D = - 2$

$9 = A + 3 - 2 D$

$C = 1$

$- 5 = B + 3 D$

Step 3.4

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

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Step 3.4.1

Replace all occurrences of $D$ in $9 = A + 3 - 2 D$ with $- 2$ .

$9 = A + 3 - 2 \cdot - 2$

$D = - 2$

$C = 1$

$- 5 = B + 3 D$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $A + 3 - 2 \cdot - 2$ .

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Step 3.4.2.1.1

Multiply $- 2$ by $- 2$ .

$9 = A + 3 + 4$

$D = - 2$

$C = 1$

$- 5 = B + 3 D$

Step 3.4.2.1.2

Add $3$ and $4$ .

$9 = A + 7$

$D = - 2$

$C = 1$

$- 5 = B + 3 D$

$9 = A + 7$

$D = - 2$

$C = 1$

$- 5 = B + 3 D$

$9 = A + 7$

$D = - 2$

$C = 1$

$- 5 = B + 3 D$

Step 3.4.3

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

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

$9 = A + 7$

$D = - 2$

$C = 1$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Multiply $3$ by $- 2$ .

$- 5 = B - 6$

$9 = A + 7$

$D = - 2$

$C = 1$

$- 5 = B - 6$

$9 = A + 7$

$D = - 2$

$C = 1$

$- 5 = B - 6$

$9 = A + 7$

$D = - 2$

$C = 1$

Step 3.5

Solve for $B$ in $- 5 = B - 6$ .

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Step 3.5.1

Rewrite the equation as $B - 6 = - 5$ .

$B - 6 = - 5$

$9 = A + 7$

$D = - 2$

$C = 1$

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 $6$ to both sides of the equation.

$B = - 5 + 6$

$9 = A + 7$

$D = - 2$

$C = 1$

Step 3.5.2.2

Add $- 5$ and $6$ .

$B = 1$

$9 = A + 7$

$D = - 2$

$C = 1$

$B = 1$

$9 = A + 7$

$D = - 2$

$C = 1$

$B = 1$

$9 = A + 7$

$D = - 2$

$C = 1$

Step 3.6

Replace all occurrences of $B$ with $1$ in each equation.

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Step 3.6.1

Rewrite the equation as $A + 7 = 9$ .

$A + 7 = 9$

$B = 1$

$D = - 2$

$C = 1$

Step 3.6.2

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

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Step 3.6.2.1

Subtract $7$ from both sides of the equation.

$A = 9 - 7$

$B = 1$

$D = - 2$

$C = 1$

Step 3.6.2.2

Subtract $7$ from $9$ .

$A = 2$

$B = 1$

$D = - 2$

$C = 1$

$A = 2$

$B = 1$

$D = - 2$

$C = 1$

$A = 2$

$B = 1$

$D = - 2$

$C = 1$

Step 3.7

List all of the solutions.

$A = 2, B = 1, D = - 2, C = 1$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Multiply $2$ by $x$ .

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

Step 5.2

Multiply $x$ by $1$ .

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