Algebra Examples

$\frac{5}{x} + \frac{x}{x + 4} = \frac{16}{x^{2} + 4 x}$

Step 1

Subtract $\frac{16}{x^{2} + 4 x}$ from both sides of the equation.

$\frac{5}{x} + \frac{x}{x + 4} - \frac{16}{x^{2} + 4 x} = 0$

Step 2

Simplify $\frac{5}{x} + \frac{x}{x + 4} - \frac{16}{x^{2} + 4 x}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{5}{x}$ by $\frac{(x + 4) (x^{2} + 4 x)}{(x + 4) (x^{2} + 4 x)}$ .

$\frac{5}{x} \cdot \frac{(x + 4) (x^{2} + 4 x)}{(x + 4) (x^{2} + 4 x)} + \frac{x}{x + 4} - \frac{16}{x^{2} + 4 x} = 0$

Step 2.1.2

Multiply $\frac{5}{x}$ by $\frac{(x + 4) (x^{2} + 4 x)}{(x + 4) (x^{2} + 4 x)}$ .

$\frac{5 ((x + 4) (x^{2} + 4 x))}{x ((x + 4) (x^{2} + 4 x))} + \frac{x}{x + 4} - \frac{16}{x^{2} + 4 x} = 0$

Step 2.1.3

Multiply $\frac{x}{x + 4}$ by $\frac{x (x^{2} + 4 x)}{x (x^{2} + 4 x)}$ .

$\frac{5 ((x + 4) (x^{2} + 4 x))}{x ((x + 4) (x^{2} + 4 x))} + \frac{x}{x + 4} \cdot \frac{x (x^{2} + 4 x)}{x (x^{2} + 4 x)} - \frac{16}{x^{2} + 4 x} = 0$

Step 2.1.4

Multiply $\frac{x}{x + 4}$ by $\frac{x (x^{2} + 4 x)}{x (x^{2} + 4 x)}$ .

$\frac{5 ((x + 4) (x^{2} + 4 x))}{x ((x + 4) (x^{2} + 4 x))} + \frac{x (x (x^{2} + 4 x))}{(x + 4) (x (x^{2} + 4 x))} - \frac{16}{x^{2} + 4 x} = 0$

Step 2.1.5

Multiply $\frac{16}{x^{2} + 4 x}$ by $\frac{x (x + 4)}{x (x + 4)}$ .

$\frac{5 ((x + 4) (x^{2} + 4 x))}{x ((x + 4) (x^{2} + 4 x))} + \frac{x (x (x^{2} + 4 x))}{(x + 4) (x (x^{2} + 4 x))} - (\frac{16}{x^{2} + 4 x} \cdot \frac{x (x + 4)}{x (x + 4)}) = 0$

Step 2.1.6

Multiply $\frac{16}{x^{2} + 4 x}$ by $\frac{x (x + 4)}{x (x + 4)}$ .

$\frac{5 ((x + 4) (x^{2} + 4 x))}{x ((x + 4) (x^{2} + 4 x))} + \frac{x (x (x^{2} + 4 x))}{(x + 4) (x (x^{2} + 4 x))} - \frac{16 (x (x + 4))}{(x^{2} + 4 x) (x (x + 4))} = 0$

Step 2.1.7

Reorder the factors of $x ((x + 4) (x^{2} + 4 x))$ .

$\frac{5 ((x + 4) (x^{2} + 4 x))}{x (x + 4) (x^{2} + 4 x)} + \frac{x (x (x^{2} + 4 x))}{(x + 4) (x (x^{2} + 4 x))} - \frac{16 (x (x + 4))}{(x^{2} + 4 x) (x (x + 4))} = 0$

Step 2.1.8

Reorder the factors of $(x + 4) (x (x^{2} + 4 x))$ .

$\frac{5 ((x + 4) (x^{2} + 4 x))}{x (x + 4) (x^{2} + 4 x)} + \frac{x (x (x^{2} + 4 x))}{x (x + 4) (x^{2} + 4 x)} - \frac{16 (x (x + 4))}{(x^{2} + 4 x) (x (x + 4))} = 0$

Step 2.1.9

Reorder the factors of $(x^{2} + 4 x) (x (x + 4))$ .

$\frac{5 ((x + 4) (x^{2} + 4 x))}{x (x + 4) (x^{2} + 4 x)} + \frac{x (x (x^{2} + 4 x))}{x (x + 4) (x^{2} + 4 x)} - \frac{16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{5 ((x + 4) (x^{2} + 4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Expand $(x + 4) (x^{2} + 4 x)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$\frac{5 (x (x^{2} + 4 x) + 4 (x^{2} + 4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.1.2

Apply the distributive property.

$\frac{5 (x \cdot x^{2} + x (4 x) + 4 (x^{2} + 4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.1.3

Apply the distributive property.

$\frac{5 (x \cdot x^{2} + x (4 x) + 4 x^{2} + 4 (4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.2

Simplify and combine like terms.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

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

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Step 2.3.2.1.1.1

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

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Step 2.3.2.1.1.1.1

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

$\frac{5 (x^{1} x^{2} + x (4 x) + 4 x^{2} + 4 (4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.2.1.1.1.2

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

$\frac{5 (x^{1 + 2} + x (4 x) + 4 x^{2} + 4 (4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.2.1.1.2

Add $1$ and $2$ .

$\frac{5 (x^{3} + x (4 x) + 4 x^{2} + 4 (4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.2.1.2

Rewrite using the commutative property of multiplication.

$\frac{5 (x^{3} + 4 x \cdot x + 4 x^{2} + 4 (4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.2.1.3

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

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Step 2.3.2.1.3.1

Move $x$ .

$\frac{5 (x^{3} + 4 (x \cdot x) + 4 x^{2} + 4 (4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.2.1.3.2

Multiply $x$ by $x$ .

$\frac{5 (x^{3} + 4 x^{2} + 4 x^{2} + 4 (4 x)) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.2.1.4

Multiply $4$ by $4$ .

$\frac{5 (x^{3} + 4 x^{2} + 4 x^{2} + 16 x) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.2.2

Add $4 x^{2}$ and $4 x^{2}$ .

$\frac{5 (x^{3} + 8 x^{2} + 16 x) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.3

Apply the distributive property.

$\frac{5 x^{3} + 5 (8 x^{2}) + 5 (16 x) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Multiply $8$ by $5$ .

$\frac{5 x^{3} + 40 x^{2} + 5 (16 x) + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.4.2

Multiply $16$ by $5$ .

$\frac{5 x^{3} + 40 x^{2} + 80 x + x (x (x^{2} + 4 x)) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.5

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

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Step 2.3.5.1

Move $x$ .

$\frac{5 x^{3} + 40 x^{2} + 80 x + x \cdot x (x^{2} + 4 x) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.5.2

Multiply $x$ by $x$ .

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{2} (x^{2} + 4 x) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.6

Apply the distributive property.

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{2} x^{2} + x^{2} (4 x) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.7

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

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Step 2.3.7.1

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

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{2 + 2} + x^{2} (4 x) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.7.2

Add $2$ and $2$ .

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + x^{2} (4 x) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.8

Rewrite using the commutative property of multiplication.

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 x^{2} x - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.9

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

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Step 2.3.9.1

Move $x$ .

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 (x \cdot x^{2}) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.9.2

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

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Step 2.3.9.2.1

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

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 (x^{1} x^{2}) - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.9.2.2

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

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 x^{1 + 2} - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.9.3

Add $1$ and $2$ .

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 x^{3} - 16 (x (x + 4))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.10

Apply the distributive property.

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 x^{3} - 16 (x \cdot x + x \cdot 4)}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.11

Multiply $x$ by $x$ .

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 x^{3} - 16 (x^{2} + x \cdot 4)}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.12

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

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 x^{3} - 16 (x^{2} + 4 \cdot x)}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.13

Apply the distributive property.

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 x^{3} - 16 x^{2} - 16 (4 x)}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.3.14

Multiply $4$ by $- 16$ .

$\frac{5 x^{3} + 40 x^{2} + 80 x + x^{4} + 4 x^{3} - 16 x^{2} - 64 x}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.4

Add $5 x^{3}$ and $4 x^{3}$ .

$\frac{9 x^{3} + 40 x^{2} + 80 x + x^{4} - 16 x^{2} - 64 x}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.5

Subtract $16 x^{2}$ from $40 x^{2}$ .

$\frac{9 x^{3} + 24 x^{2} + 80 x + x^{4} - 64 x}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.6

Subtract $64 x$ from $80 x$ .

$\frac{9 x^{3} + 24 x^{2} + x^{4} + 16 x}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Factor $x$ out of $9 x^{3} + 24 x^{2} + x^{4} + 16 x$ .

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Step 2.7.1.1

Factor $x$ out of $9 x^{3}$ .

$\frac{x (9 x^{2}) + 24 x^{2} + x^{4} + 16 x}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.1.2

Factor $x$ out of $24 x^{2}$ .

$\frac{x (9 x^{2}) + x (24 x) + x^{4} + 16 x}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.1.3

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

$\frac{x (9 x^{2}) + x (24 x) + x \cdot x^{3} + 16 x}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.1.4

Factor $x$ out of $16 x$ .

$\frac{x (9 x^{2}) + x (24 x) + x \cdot x^{3} + x \cdot 16}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.1.5

Factor $x$ out of $x (9 x^{2}) + x (24 x)$ .

$\frac{x (9 x^{2} + 24 x) + x \cdot x^{3} + x \cdot 16}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.1.6

Factor $x$ out of $x (9 x^{2} + 24 x) + x \cdot x^{3}$ .

$\frac{x (9 x^{2} + 24 x + x^{3}) + x \cdot 16}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.1.7

Factor $x$ out of $x (9 x^{2} + 24 x + x^{3}) + x \cdot 16$ .

$\frac{x (9 x^{2} + 24 x + x^{3} + 16)}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.2

Reorder terms.

$\frac{x (x^{3} + 9 x^{2} + 24 x + 16)}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.3

Rewrite $x^{3} + 9 x^{2} + 24 x + 16$ in a factored form.

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Step 2.7.3.1

Factor $x^{3} + 9 x^{2} + 24 x + 16$ using the rational roots test.

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Step 2.7.3.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

$q = \pm 1$

Step 2.7.3.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 16, \pm 2, \pm 8, \pm 4$

Step 2.7.3.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 2.7.3.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} + 9 {(- 1)}^{2} + 24 \cdot - 1 + 16$

Step 2.7.3.1.3.2

Raise $- 1$ to the power of $3$ .

$- 1 + 9 {(- 1)}^{2} + 24 \cdot - 1 + 16$

Step 2.7.3.1.3.3

Raise $- 1$ to the power of $2$ .

$- 1 + 9 \cdot 1 + 24 \cdot - 1 + 16$

Step 2.7.3.1.3.4

Multiply $9$ by $1$ .

$- 1 + 9 + 24 \cdot - 1 + 16$

Step 2.7.3.1.3.5

Add $- 1$ and $9$ .

$8 + 24 \cdot - 1 + 16$

Step 2.7.3.1.3.6

Multiply $24$ by $- 1$ .

$8 - 24 + 16$

Step 2.7.3.1.3.7

Subtract $24$ from $8$ .

$- 16 + 16$

Step 2.7.3.1.3.8

Add $- 16$ and $16$ .

$0$

Step 2.7.3.1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 9 x^{2} + 24 x + 16}{x + 1}$

Step 2.7.3.1.5

Divide $x^{3} + 9 x^{2} + 24 x + 16$ by $x + 1$ .

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Step 2.7.3.1.5.1

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

$x$

$1$

$x^{3}$

$9 x^{2}$

$24 x$

$16$

Step 2.7.3.1.5.2

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$

Step 2.7.3.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			+	$x^{3}$	+	$x^{2}$

Step 2.7.3.1.5.4

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$

Step 2.7.3.1.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$

Step 2.7.3.1.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$

Step 2.7.3.1.5.7

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$

Step 2.7.3.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$
					+	$8 x^{2}$	+	$8 x$

Step 2.7.3.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} + 8 x$

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$
					-	$8 x^{2}$	-	$8 x$

Step 2.7.3.1.5.10

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

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$
					-	$8 x^{2}$	-	$8 x$
							+	$16 x$

Step 2.7.3.1.5.11

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

				$x^{2}$	+	$8 x$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$
					-	$8 x^{2}$	-	$8 x$
							+	$16 x$	+	$16$

Step 2.7.3.1.5.12

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$8 x$	+	$16$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$
					-	$8 x^{2}$	-	$8 x$
							+	$16 x$	+	$16$

Step 2.7.3.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$	+	$16$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$
					-	$8 x^{2}$	-	$8 x$
							+	$16 x$	+	$16$
							+	$16 x$	+	$16$

Step 2.7.3.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x + 16$

				$x^{2}$	+	$8 x$	+	$16$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$
					-	$8 x^{2}$	-	$8 x$
							+	$16 x$	+	$16$
							-	$16 x$	-	$16$

Step 2.7.3.1.5.15

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

				$x^{2}$	+	$8 x$	+	$16$
$x$	+	$1$		$x^{3}$	+	$9 x^{2}$	+	$24 x$	+	$16$
			-	$x^{3}$	-	$x^{2}$
					+	$8 x^{2}$	+	$24 x$
					-	$8 x^{2}$	-	$8 x$
							+	$16 x$	+	$16$
							-	$16 x$	-	$16$
										$0$

Step 2.7.3.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 8 x + 16$

Step 2.7.3.1.6

Write $x^{3} + 9 x^{2} + 24 x + 16$ as a set of factors.

$\frac{x ((x + 1) (x^{2} + 8 x + 16))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.3.2

Factor using the perfect square rule.

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Step 2.7.3.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{x ((x + 1) (x^{2} + 8 x + 4^{2}))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.3.2.2

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

$8 x = 2 \cdot x \cdot 4$

Step 2.7.3.2.3

Rewrite the polynomial.

$\frac{x ((x + 1) (x^{2} + 2 \cdot x \cdot 4 + 4^{2}))}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.7.3.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$\frac{x ((x + 1) {(x + 4)}^{2})}{x (x + 4) (x^{2} + 4 x)} = 0$

$\frac{x (x + 1) {(x + 4)}^{2}}{x (x + 4) (x^{2} + 4 x)} = 0$

Step 2.8

Simplify the denominator.

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Step 2.8.1

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

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Step 2.8.1.1

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

$\frac{x (x + 1) {(x + 4)}^{2}}{x (x + 4) (x \cdot x + 4 x)} = 0$

Step 2.8.1.2

Factor $x$ out of $4 x$ .

$\frac{x (x + 1) {(x + 4)}^{2}}{x (x + 4) (x \cdot x + x \cdot 4)} = 0$

Step 2.8.1.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$\frac{x (x + 1) {(x + 4)}^{2}}{x (x + 4) (x (x + 4))} = 0$

$\frac{x (x + 1) {(x + 4)}^{2}}{x (x + 4) x (x + 4)} = 0$

Step 2.8.2

Combine exponents.

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Step 2.8.2.1

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

$\frac{x (x + 1) {(x + 4)}^{2}}{x^{1} x (x + 4) (x + 4)} = 0$

Step 2.8.2.2

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

$\frac{x (x + 1) {(x + 4)}^{2}}{x^{1} x^{1} (x + 4) (x + 4)} = 0$

Step 2.8.2.3

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

$\frac{x (x + 1) {(x + 4)}^{2}}{x^{1 + 1} (x + 4) (x + 4)} = 0$

Step 2.8.2.4

Add $1$ and $1$ .

$\frac{x (x + 1) {(x + 4)}^{2}}{x^{2} (x + 4) (x + 4)} = 0$

Step 2.8.2.5

Raise $x + 4$ to the power of $1$ .

$\frac{x (x + 1) {(x + 4)}^{2}}{x^{2} ({(x + 4)}^{1} (x + 4))} = 0$

Step 2.8.2.6

Raise $x + 4$ to the power of $1$ .

$\frac{x (x + 1) {(x + 4)}^{2}}{x^{2} ({(x + 4)}^{1} {(x + 4)}^{1})} = 0$

Step 2.8.2.7

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

$\frac{x (x + 1) {(x + 4)}^{2}}{x^{2} {(x + 4)}^{1 + 1}} = 0$

Step 2.8.2.8

Add $1$ and $1$ .

$\frac{x (x + 1) {(x + 4)}^{2}}{x^{2} {(x + 4)}^{2}} = 0$

Step 2.9

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

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Step 2.9.1

Factor $x$ out of $x (x + 1) {(x + 4)}^{2}$ .

$\frac{x ((x + 1) {(x + 4)}^{2})}{x^{2} {(x + 4)}^{2}} = 0$

Step 2.9.2

Cancel the common factors.

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Step 2.9.2.1

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

$\frac{x ((x + 1) {(x + 4)}^{2})}{x (x {(x + 4)}^{2})} = 0$

Step 2.9.2.2

Cancel the common factor.

$\frac{x ((x + 1) {(x + 4)}^{2})}{x (x {(x + 4)}^{2})} = 0$

Step 2.9.2.3

Rewrite the expression.

$\frac{(x + 1) {(x + 4)}^{2}}{x {(x + 4)}^{2}} = 0$

Step 2.10

Cancel the common factor of ${(x + 4)}^{2}$ .

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Step 2.10.1

Cancel the common factor.

$\frac{(x + 1) {(x + 4)}^{2}}{x {(x + 4)}^{2}} = 0$

Step 2.10.2

Rewrite the expression.

$\frac{x + 1}{x} = 0$

Step 3

Set the numerator equal to zero.

$x + 1 = 0$

Step 4

Subtract $1$ from both sides of the equation.

$x = - 1$