Algebra Examples

$\frac{4 x^{5} + 17 x^{4} - 16 x^{3} - 5 x^{2} + x}{4 x^{2} + 21 x + 5}$

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 $0$ .

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

Step 1.2

Divide the highest order term in the dividend $4 x^{5}$ by the highest order term in divisor $4 x^{2}$ .

$x^{3}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

$4 x^{5}$

$21 x^{4}$

$5 x^{3}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{5} + 21 x^{4} + 5 x^{3}$

$x^{3}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

$4 x^{5}$

$21 x^{4}$

$5 x^{3}$

Step 1.5

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

$x^{3}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

$4 x^{5}$

$21 x^{4}$

$5 x^{3}$

$4 x^{4}$

$21 x^{3}$

Step 1.6

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

$x^{3}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

$4 x^{5}$

$21 x^{4}$

$5 x^{3}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

Step 1.7

Divide the highest order term in the dividend $- 4 x^{4}$ by the highest order term in divisor $4 x^{2}$ .

$x^{3}$

$x^{2}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

$4 x^{5}$

$21 x^{4}$

$5 x^{3}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

Step 1.8

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

$4 x^{5}$

$21 x^{4}$

$5 x^{3}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{4} - 21 x^{3} - 5 x^{2}$

$x^{3}$

$x^{2}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

$4 x^{5}$

$21 x^{4}$

$5 x^{3}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

Step 1.10

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

$x^{3}$

$x^{2}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

$4 x^{5}$

$21 x^{4}$

$5 x^{3}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

$0$

Step 1.11

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

$x^{3}$

$x^{2}$

$4 x^{2}$

$21 x$

$5$

$4 x^{5}$

$17 x^{4}$

$16 x^{3}$

$5 x^{2}$

$x$

$0$

$4 x^{5}$

$21 x^{4}$

$5 x^{3}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

$4 x^{4}$

$21 x^{3}$

$5 x^{2}$

$0$

$x$

$0$

Step 1.12

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

$x^{3} - x^{2} + \frac{x}{4 x^{2} + 21 x + 5}$

Step 2

Decompose the fraction and multiply through by the common denominator.

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Step 2.1

Factor by grouping.

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Step 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 $a \cdot c = 4 \cdot 5 = 20$ and whose sum is $b = 21$ .

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Step 2.1.1.1

Factor $21$ out of $21 x$ .

$\frac{x}{4 x^{2} + 21 (x) + 5}$

Step 2.1.1.2

Rewrite $21$ as $1$ plus $20$

$\frac{x}{4 x^{2} + (1 + 20) x + 5}$

Step 2.1.1.3

Apply the distributive property.

$\frac{x}{4 x^{2} + 1 x + 20 x + 5}$

Step 2.1.1.4

Multiply $x$ by $1$ .

$\frac{x}{4 x^{2} + x + 20 x + 5}$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$\frac{x}{(4 x^{2} + x) + 20 x + 5}$

Step 2.1.2.2

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

$\frac{x}{x (4 x + 1) + 5 (4 x + 1)}$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$\frac{x}{(4 x + 1) (x + 5)}$

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 $A$ .

$\frac{A}{4 x + 1}$

Step 2.3

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

Step 2.4

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

$\frac{x (4 x + 1) (x + 5)}{(4 x + 1) (x + 5)} = \frac{(A) (4 x + 1) (x + 5)}{4 x + 1} + \frac{(B) (4 x + 1) (x + 5)}{x + 5}$

Step 2.5

Cancel the common factor of $4 x + 1$ .

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Step 2.5.1

Cancel the common factor.

$\frac{x (4 x + 1) (x + 5)}{(4 x + 1) (x + 5)} = \frac{(A) (4 x + 1) (x + 5)}{4 x + 1} + \frac{(B) (4 x + 1) (x + 5)}{x + 5}$

Step 2.5.2

Rewrite the expression.

$\frac{x (x + 5)}{x + 5} = \frac{(A) (4 x + 1) (x + 5)}{4 x + 1} + \frac{(B) (4 x + 1) (x + 5)}{x + 5}$

Step 2.6

Cancel the common factor of $x + 5$ .

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Step 2.6.1

Cancel the common factor.

$\frac{x (x + 5)}{x + 5} = \frac{(A) (4 x + 1) (x + 5)}{4 x + 1} + \frac{(B) (4 x + 1) (x + 5)}{x + 5}$

Step 2.6.2

Divide $x$ by $1$ .

$x = \frac{(A) (4 x + 1) (x + 5)}{4 x + 1} + \frac{(B) (4 x + 1) (x + 5)}{x + 5}$

Step 2.7

Simplify each term.

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Step 2.7.1

Cancel the common factor of $4 x + 1$ .

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Step 2.7.1.1

Cancel the common factor.

$x = \frac{A (4 x + 1) (x + 5)}{4 x + 1} + \frac{(B) (4 x + 1) (x + 5)}{x + 5}$

Step 2.7.1.2

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

$x = (A) (x + 5) + \frac{(B) (4 x + 1) (x + 5)}{x + 5}$

Step 2.7.2

Apply the distributive property.

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

Step 2.7.3

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

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

Step 2.7.4

Cancel the common factor of $x + 5$ .

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Step 2.7.4.1

Cancel the common factor.

$x = A x + 5 A + \frac{(B) (4 x + 1) (x + 5)}{x + 5}$

Step 2.7.4.2

Divide $(B) (4 x + 1)$ by $1$ .

$x = A x + 5 A + (B) (4 x + 1)$

Step 2.7.5

Apply the distributive property.

$x = A x + 5 A + B (4 x) + B \cdot 1$

Step 2.7.6

Rewrite using the commutative property of multiplication.

$x = A x + 5 A + 4 B x + B \cdot 1$

Step 2.7.7

Multiply $B$ by $1$ .

$x = A x + 5 A + 4 B x + B$

Step 2.8

Simplify the expression.

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Step 2.8.1

Move $B$ .

$x = A x + 5 A + 4 x B + B$

Step 2.8.2

Move $5 A$ .

$x = A x + 4 x B + 5 A + B$

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.

$1 = A + 4 B$

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.

$0 = 5 A + B$

Step 3.3

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

$1 = A + 4 B$

$0 = 5 A + B$

$1 = A + 4 B$

$0 = 5 A + B$

Step 4

Solve the system of equations.

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Step 4.1

Solve for $A$ in $1 = A + 4 B$ .

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Step 4.1.1

Rewrite the equation as $A + 4 B = 1$ .

$A + 4 B = 1$

$0 = 5 A + B$

Step 4.1.2

Subtract $4 B$ from both sides of the equation.

$A = 1 - 4 B$

$0 = 5 A + B$

$A = 1 - 4 B$

$0 = 5 A + B$

Step 4.2

Replace all occurrences of $A$ with $1 - 4 B$ in each equation.

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Step 4.2.1

Replace all occurrences of $A$ in $0 = 5 A + B$ with $1 - 4 B$ .

$0 = 5 (1 - 4 B) + B$

$A = 1 - 4 B$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $5 (1 - 4 B) + B$ .

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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.

$0 = 5 \cdot 1 + 5 (- 4 B) + B$

$A = 1 - 4 B$

Step 4.2.2.1.1.2

Multiply $5$ by $1$ .

$0 = 5 + 5 (- 4 B) + B$

$A = 1 - 4 B$

Step 4.2.2.1.1.3

Multiply $- 4$ by $5$ .

$0 = 5 - 20 B + B$

$A = 1 - 4 B$

$0 = 5 - 20 B + B$

$A = 1 - 4 B$

Step 4.2.2.1.2

Add $- 20 B$ and $B$ .

$0 = 5 - 19 B$

$A = 1 - 4 B$

$0 = 5 - 19 B$

$A = 1 - 4 B$

$0 = 5 - 19 B$

$A = 1 - 4 B$

$0 = 5 - 19 B$

$A = 1 - 4 B$

Step 4.3

Solve for $B$ in $0 = 5 - 19 B$ .

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Step 4.3.1

Rewrite the equation as $5 - 19 B = 0$ .

$5 - 19 B = 0$

$A = 1 - 4 B$

Step 4.3.2

Subtract $5$ from both sides of the equation.

$- 19 B = - 5$

$A = 1 - 4 B$

Step 4.3.3

Divide each term in $- 19 B = - 5$ by $- 19$ and simplify.

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Step 4.3.3.1

Divide each term in $- 19 B = - 5$ by $- 19$ .

$\frac{- 19 B}{- 19} = \frac{- 5}{- 19}$

$A = 1 - 4 B$

Step 4.3.3.2

Simplify the left side.

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Step 4.3.3.2.1

Cancel the common factor of $- 19$ .

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Step 4.3.3.2.1.1

Cancel the common factor.

$\frac{- 19 B}{- 19} = \frac{- 5}{- 19}$

$A = 1 - 4 B$

Step 4.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 5}{- 19}$

$A = 1 - 4 B$

$B = \frac{- 5}{- 19}$

$A = 1 - 4 B$

$B = \frac{- 5}{- 19}$

$A = 1 - 4 B$

Step 4.3.3.3

Simplify the right side.

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Step 4.3.3.3.1

Dividing two negative values results in a positive value.

$B = \frac{5}{19}$

$A = 1 - 4 B$

$B = \frac{5}{19}$

$A = 1 - 4 B$

$B = \frac{5}{19}$

$A = 1 - 4 B$

$B = \frac{5}{19}$

$A = 1 - 4 B$

Step 4.4

Replace all occurrences of $B$ with $\frac{5}{19}$ in each equation.

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Step 4.4.1

Replace all occurrences of $B$ in $A = 1 - 4 B$ with $\frac{5}{19}$ .

$A = 1 - 4 (\frac{5}{19})$

$B = \frac{5}{19}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $1 - 4 (\frac{5}{19})$ .

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Step 4.4.2.1.1

Simplify each term.

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Step 4.4.2.1.1.1

Multiply $- 4 (\frac{5}{19})$ .

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Step 4.4.2.1.1.1.1

Combine $- 4$ and $\frac{5}{19}$ .

$A = 1 + \frac{- 4 \cdot 5}{19}$

$B = \frac{5}{19}$

Step 4.4.2.1.1.1.2

Multiply $- 4$ by $5$ .

$A = 1 + \frac{- 20}{19}$

$B = \frac{5}{19}$

$A = 1 + \frac{- 20}{19}$

$B = \frac{5}{19}$

Step 4.4.2.1.1.2

Move the negative in front of the fraction.

$A = 1 - \frac{20}{19}$

$B = \frac{5}{19}$

$A = 1 - \frac{20}{19}$

$B = \frac{5}{19}$

Step 4.4.2.1.2

Simplify the expression.

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Step 4.4.2.1.2.1

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

$A = \frac{19}{19} - \frac{20}{19}$

$B = \frac{5}{19}$

Step 4.4.2.1.2.2

Combine the numerators over the common denominator.

$A = \frac{19 - 20}{19}$

$B = \frac{5}{19}$

Step 4.4.2.1.2.3

Subtract $20$ from $19$ .

$A = \frac{- 1}{19}$

$B = \frac{5}{19}$

Step 4.4.2.1.2.4

Move the negative in front of the fraction.

$A = - \frac{1}{19}$

$B = \frac{5}{19}$

$A = - \frac{1}{19}$

$B = \frac{5}{19}$

$A = - \frac{1}{19}$

$B = \frac{5}{19}$

$A = - \frac{1}{19}$

$B = \frac{5}{19}$

$A = - \frac{1}{19}$

$B = \frac{5}{19}$

Step 4.5

List all of the solutions.

$A = - \frac{1}{19}, B = \frac{5}{19}$

Step 5

Replace each of the partial fraction coefficients in $x^{3} - x^{2} + \frac{A}{4 x + 1} + \frac{B}{x + 5}$ with the values found for $A$ and $B$ .

$x^{3} - x^{2} + \frac{- \frac{1}{19}}{4 x + 1} + \frac{\frac{5}{19}}{x + 5}$