Algebra Examples

$f (x) = x^{3} + 12 x^{2} + 25 x - 70$ $g (x) = x + 7$

Step 1

Substitute $x + 7$ for $f (x)$ .

$x + 7 = x^{3} + 12 x^{2} + 25 x - 70$

Step 2

Solve for $x$ .

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Step 2.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{3} + 12 x^{2} + 25 x - 70 = x + 7$

Step 2.2

Move all terms containing $x$ to the left side of the equation.

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Step 2.2.1

Subtract $x$ from both sides of the equation.

$x^{3} + 12 x^{2} + 25 x - 70 - x = 7$

Step 2.2.2

Subtract $x$ from $25 x$ .

$x^{3} + 12 x^{2} + 24 x - 70 = 7$

Step 2.3

Subtract $7$ from both sides of the equation.

$x^{3} + 12 x^{2} + 24 x - 70 - 7 = 0$

Step 2.4

Subtract $7$ from $- 70$ .

$x^{3} + 12 x^{2} + 24 x - 77 = 0$

Step 2.5

Factor $x^{3} + 12 x^{2} + 24 x - 77$ using the rational roots test.

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Step 2.5.1

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.

$p = \pm 1, \pm 77, \pm 7, \pm 11$

$q = \pm 1$

Step 2.5.2

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

$\pm 1, \pm 77, \pm 7, \pm 11$

Step 2.5.3

Substitute $- 7$ and simplify the expression. In this case, the expression is equal to $0$ so $- 7$ is a root of the polynomial.

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Step 2.5.3.1

Substitute $- 7$ into the polynomial.

${(- 7)}^{3} + 12 {(- 7)}^{2} + 24 \cdot - 7 - 77$

Step 2.5.3.2

Raise $- 7$ to the power of $3$ .

$- 343 + 12 {(- 7)}^{2} + 24 \cdot - 7 - 77$

Step 2.5.3.3

Raise $- 7$ to the power of $2$ .

$- 343 + 12 \cdot 49 + 24 \cdot - 7 - 77$

Step 2.5.3.4

Multiply $12$ by $49$ .

$- 343 + 588 + 24 \cdot - 7 - 77$

Step 2.5.3.5

Add $- 343$ and $588$ .

$245 + 24 \cdot - 7 - 77$

Step 2.5.3.6

Multiply $24$ by $- 7$ .

$245 - 168 - 77$

Step 2.5.3.7

Subtract $168$ from $245$ .

$77 - 77$

Step 2.5.3.8

Subtract $77$ from $77$ .

$0$

Step 2.5.4

Since $- 7$ is a known root, divide the polynomial by $x + 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 12 x^{2} + 24 x - 77}{x + 7}$

Step 2.5.5

Divide $x^{3} + 12 x^{2} + 24 x - 77$ by $x + 7$ .

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Step 2.5.5.1

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

$x$

$7$

$x^{3}$

$12 x^{2}$

$24 x$

$77$

Step 2.5.5.2

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

					$x^{2}$
	$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$

Step 2.5.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			+	$x^{3}$	+	$7 x^{2}$

Step 2.5.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 7 x^{2}$

				$x^{2}$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$

Step 2.5.5.5

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

				$x^{2}$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$

Step 2.5.5.6

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

				$x^{2}$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$

Step 2.5.5.7

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

				$x^{2}$	+	$5 x$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$

Step 2.5.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$5 x$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$
					+	$5 x^{2}$	+	$35 x$

Step 2.5.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 35 x$

				$x^{2}$	+	$5 x$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$
					-	$5 x^{2}$	-	$35 x$

Step 2.5.5.10

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

				$x^{2}$	+	$5 x$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$
					-	$5 x^{2}$	-	$35 x$
							-	$11 x$

Step 2.5.5.11

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

				$x^{2}$	+	$5 x$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$
					-	$5 x^{2}$	-	$35 x$
							-	$11 x$	-	$77$

Step 2.5.5.12

Divide the highest order term in the dividend $- 11 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$5 x$	-	$11$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$
					-	$5 x^{2}$	-	$35 x$
							-	$11 x$	-	$77$

Step 2.5.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$5 x$	-	$11$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$
					-	$5 x^{2}$	-	$35 x$
							-	$11 x$	-	$77$
							-	$11 x$	-	$77$

Step 2.5.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 11 x - 77$

				$x^{2}$	+	$5 x$	-	$11$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$
					-	$5 x^{2}$	-	$35 x$
							-	$11 x$	-	$77$
							+	$11 x$	+	$77$

Step 2.5.5.15

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

				$x^{2}$	+	$5 x$	-	$11$
$x$	+	$7$		$x^{3}$	+	$12 x^{2}$	+	$24 x$	-	$77$
			-	$x^{3}$	-	$7 x^{2}$
					+	$5 x^{2}$	+	$24 x$
					-	$5 x^{2}$	-	$35 x$
							-	$11 x$	-	$77$
							+	$11 x$	+	$77$
										$0$

Step 2.5.5.16

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

$x^{2} + 5 x - 11$

Step 2.5.6

Write $x^{3} + 12 x^{2} + 24 x - 77$ as a set of factors.

$(x + 7) (x^{2} + 5 x - 11) = 0$

Step 2.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 7 = 0$

$x^{2} + 5 x - 11 = 0$

Step 2.7

Set $x + 7$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 2.7.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 2.8

Set $x^{2} + 5 x - 11$ equal to $0$ and solve for $x$ .

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Step 2.8.1

Set $x^{2} + 5 x - 11$ equal to $0$ .

$x^{2} + 5 x - 11 = 0$

Step 2.8.2

Solve $x^{2} + 5 x - 11 = 0$ for $x$ .

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Step 2.8.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.8.2.2

Substitute the values $a = 1$ , $b = 5$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 11)}}{2 \cdot 1}$

Step 2.8.2.3

Simplify.

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Step 2.8.2.3.1

Simplify the numerator.

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Step 2.8.2.3.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Step 2.8.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 11$ .

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Step 2.8.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 11}}{2 \cdot 1}$

Step 2.8.2.3.1.2.2

Multiply $- 4$ by $- 11$ .

$x = \frac{- 5 \pm \sqrt{25 + 44}}{2 \cdot 1}$

Step 2.8.2.3.1.3

Add $25$ and $44$ .

$x = \frac{- 5 \pm \sqrt{69}}{2 \cdot 1}$

Step 2.8.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{69}}{2}$

Step 2.8.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.8.2.4.1

Simplify the numerator.

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Step 2.8.2.4.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Step 2.8.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 11$ .

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Step 2.8.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 11}}{2 \cdot 1}$

Step 2.8.2.4.1.2.2

Multiply $- 4$ by $- 11$ .

$x = \frac{- 5 \pm \sqrt{25 + 44}}{2 \cdot 1}$

Step 2.8.2.4.1.3

Add $25$ and $44$ .

$x = \frac{- 5 \pm \sqrt{69}}{2 \cdot 1}$

Step 2.8.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{69}}{2}$

Step 2.8.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 5 + \sqrt{69}}{2}$

Step 2.8.2.4.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + \sqrt{69}}{2}$

Step 2.8.2.4.5

Factor $- 1$ out of $\sqrt{69}$ .

$x = \frac{- 1 \cdot 5 - 1 (- \sqrt{69})}{2}$

Step 2.8.2.4.6

Factor $- 1$ out of $- 1 (5) - 1 (- \sqrt{69})$ .

$x = \frac{- 1 (5 - \sqrt{69})}{2}$

Step 2.8.2.4.7

Move the negative in front of the fraction.

$x = - \frac{5 - \sqrt{69}}{2}$

Step 2.8.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.8.2.5.1

Simplify the numerator.

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Step 2.8.2.5.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Step 2.8.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 11$ .

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Step 2.8.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 11}}{2 \cdot 1}$

Step 2.8.2.5.1.2.2

Multiply $- 4$ by $- 11$ .

$x = \frac{- 5 \pm \sqrt{25 + 44}}{2 \cdot 1}$

Step 2.8.2.5.1.3

Add $25$ and $44$ .

$x = \frac{- 5 \pm \sqrt{69}}{2 \cdot 1}$

Step 2.8.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{69}}{2}$

Step 2.8.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 5 - \sqrt{69}}{2}$

Step 2.8.2.5.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - \sqrt{69}}{2}$

Step 2.8.2.5.5

Factor $- 1$ out of $- \sqrt{69}$ .

$x = \frac{- 1 \cdot 5 - (\sqrt{69})}{2}$

Step 2.8.2.5.6

Factor $- 1$ out of $- 1 (5) - (\sqrt{69})$ .

$x = \frac{- 1 (5 + \sqrt{69})}{2}$

Step 2.8.2.5.7

Move the negative in front of the fraction.

$x = - \frac{5 + \sqrt{69}}{2}$

Step 2.8.2.6

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{69}}{2}, - \frac{5 + \sqrt{69}}{2}$

Step 2.9

The final solution is all the values that make $(x + 7) (x^{2} + 5 x - 11) = 0$ true.

$x = - 7, - \frac{5 - \sqrt{69}}{2}, - \frac{5 + \sqrt{69}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - 7, - \frac{5 - \sqrt{69}}{2}, - \frac{5 + \sqrt{69}}{2}$

Decimal Form:

$x = - 7, 1.65331193 \dots, - 6.65331193 \dots$