Pre-Algebra Examples

$x^{4} - 3 x^{3} - 15 x^{2} - 17 x - 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x^{4} - 3 x^{3} - 15 x^{2} - 17 x - 6$ using the rational roots test.

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Step 1.1.1

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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

Step 1.1.2

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

$\pm 1, \pm 6, \pm 2, \pm 3$

Step 1.1.3

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 1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - 3 {(- 1)}^{3} - 15 {(- 1)}^{2} - 17 \cdot - 1 - 6$

Step 1.1.3.2

Raise $- 1$ to the power of $4$ .

$1 - 3 {(- 1)}^{3} - 15 {(- 1)}^{2} - 17 \cdot - 1 - 6$

Step 1.1.3.3

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

$1 - 3 \cdot - 1 - 15 {(- 1)}^{2} - 17 \cdot - 1 - 6$

Step 1.1.3.4

Multiply $- 3$ by $- 1$ .

$1 + 3 - 15 {(- 1)}^{2} - 17 \cdot - 1 - 6$

Step 1.1.3.5

Add $1$ and $3$ .

$4 - 15 {(- 1)}^{2} - 17 \cdot - 1 - 6$

Step 1.1.3.6

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

$4 - 15 \cdot 1 - 17 \cdot - 1 - 6$

Step 1.1.3.7

Multiply $- 15$ by $1$ .

$4 - 15 - 17 \cdot - 1 - 6$

Step 1.1.3.8

Subtract $15$ from $4$ .

$- 11 - 17 \cdot - 1 - 6$

Step 1.1.3.9

Multiply $- 17$ by $- 1$ .

$- 11 + 17 - 6$

Step 1.1.3.10

Add $- 11$ and $17$ .

$6 - 6$

Step 1.1.3.11

Subtract $6$ from $6$ .

$0$

Step 1.1.4

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.

$\frac{x^{4} - 3 x^{3} - 15 x^{2} - 17 x - 6}{x + 1}$

Step 1.1.5

Divide $x^{4} - 3 x^{3} - 15 x^{2} - 17 x - 6$ by $x + 1$ .

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Step 1.1.5.1

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

$x$

$1$

$x^{4}$

$3 x^{3}$

$15 x^{2}$

$17 x$

$6$

Step 1.1.5.2

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

					$x^{3}$
	$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$3 x^{3}$

$15 x^{2}$

$17 x$

$6$

$x^{4}$

$x^{3}$

Step 1.1.5.4

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$

Step 1.1.5.5

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$

Step 1.1.5.6

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$

Step 1.1.5.7

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x$ .

				$x^{3}$	-	$4 x^{2}$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$4 x^{2}$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					-	$4 x^{3}$	-	$4 x^{2}$

Step 1.1.5.9

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

				$x^{3}$	-	$4 x^{2}$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$

Step 1.1.5.10

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

				$x^{3}$	-	$4 x^{2}$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$
							-	$11 x^{2}$

Step 1.1.5.11

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

				$x^{3}$	-	$4 x^{2}$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$
							-	$11 x^{2}$	-	$17 x$

Step 1.1.5.12

Divide the highest order term in the dividend $- 11 x^{2}$ by the highest order term in divisor $x$ .

				$x^{3}$	-	$4 x^{2}$	-	$11 x$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$
							-	$11 x^{2}$	-	$17 x$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$4 x^{2}$	-	$11 x$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$
							-	$11 x^{2}$	-	$17 x$
							-	$11 x^{2}$	-	$11 x$

Step 1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 11 x^{2} - 11 x$

$x^{3}$

$4 x^{2}$

$11 x$

$x$

$1$

$x^{4}$

$3 x^{3}$

$15 x^{2}$

$17 x$

$6$

$x^{4}$

$x^{3}$

$4 x^{3}$

$15 x^{2}$

$4 x^{3}$

$4 x^{2}$

$11 x^{2}$

$17 x$

$11 x^{2}$

$11 x$

Step 1.1.5.15

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

$x^{3}$

$4 x^{2}$

$11 x$

$x$

$1$

$x^{4}$

$3 x^{3}$

$15 x^{2}$

$17 x$

$6$

$x^{4}$

$x^{3}$

$4 x^{3}$

$15 x^{2}$

$4 x^{3}$

$4 x^{2}$

$11 x^{2}$

$17 x$

$11 x^{2}$

$11 x$

$6 x$

Step 1.1.5.16

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

				$x^{3}$	-	$4 x^{2}$	-	$11 x$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$
							-	$11 x^{2}$	-	$17 x$
							+	$11 x^{2}$	+	$11 x$
									-	$6 x$	-	$6$

Step 1.1.5.17

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

				$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$
							-	$11 x^{2}$	-	$17 x$
							+	$11 x^{2}$	+	$11 x$
									-	$6 x$	-	$6$

Step 1.1.5.18

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$
							-	$11 x^{2}$	-	$17 x$
							+	$11 x^{2}$	+	$11 x$
									-	$6 x$	-	$6$
									-	$6 x$	-	$6$

Step 1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x - 6$

				$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$
							-	$11 x^{2}$	-	$17 x$
							+	$11 x^{2}$	+	$11 x$
									-	$6 x$	-	$6$
									+	$6 x$	+	$6$

Step 1.1.5.20

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

				$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
$x$	+	$1$		$x^{4}$	-	$3 x^{3}$	-	$15 x^{2}$	-	$17 x$	-	$6$
			-	$x^{4}$	-	$x^{3}$
					-	$4 x^{3}$	-	$15 x^{2}$
					+	$4 x^{3}$	+	$4 x^{2}$
							-	$11 x^{2}$	-	$17 x$
							+	$11 x^{2}$	+	$11 x$
									-	$6 x$	-	$6$
									+	$6 x$	+	$6$
												$0$

Step 1.1.5.21

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

$x^{3} - 4 x^{2} - 11 x - 6$

Step 1.1.6

Write $x^{4} - 3 x^{3} - 15 x^{2} - 17 x - 6$ as a set of factors.

$(x + 1) (x^{3} - 4 x^{2} - 11 x - 6) = 0$

Step 1.2

Factor $x^{3} - 4 x^{2} - 11 x - 6$ using the rational roots test.

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Step 1.2.1

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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

Step 1.2.2

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

$\pm 1, \pm 6, \pm 2, \pm 3$

Step 1.2.3

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 1.2.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 4 {(- 1)}^{2} - 11 \cdot - 1 - 6$

Step 1.2.3.2

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

$- 1 - 4 {(- 1)}^{2} - 11 \cdot - 1 - 6$

Step 1.2.3.3

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

$- 1 - 4 \cdot 1 - 11 \cdot - 1 - 6$

Step 1.2.3.4

Multiply $- 4$ by $1$ .

$- 1 - 4 - 11 \cdot - 1 - 6$

Step 1.2.3.5

Subtract $4$ from $- 1$ .

$- 5 - 11 \cdot - 1 - 6$

Step 1.2.3.6

Multiply $- 11$ by $- 1$ .

$- 5 + 11 - 6$

Step 1.2.3.7

Add $- 5$ and $11$ .

$6 - 6$

Step 1.2.3.8

Subtract $6$ from $6$ .

$0$

Step 1.2.4

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.

$\frac{x^{3} - 4 x^{2} - 11 x - 6}{x + 1}$

Step 1.2.5

Divide $x^{3} - 4 x^{2} - 11 x - 6$ by $x + 1$ .

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Step 1.2.5.1

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

$x$

$1$

$x^{3}$

$4 x^{2}$

$11 x$

$6$

Step 1.2.5.2

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$

Step 1.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			+	$x^{3}$	+	$x^{2}$

Step 1.2.5.4

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$

Step 1.2.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$

Step 1.2.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$

Step 1.2.5.7

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

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$

Step 1.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$
					-	$5 x^{2}$	-	$5 x$

Step 1.2.5.9

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

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$
					+	$5 x^{2}$	+	$5 x$

Step 1.2.5.10

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

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$
					+	$5 x^{2}$	+	$5 x$
							-	$6 x$

Step 1.2.5.11

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

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$
					+	$5 x^{2}$	+	$5 x$
							-	$6 x$	-	$6$

Step 1.2.5.12

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$	-	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$
					+	$5 x^{2}$	+	$5 x$
							-	$6 x$	-	$6$

Step 1.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$	-	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$
					+	$5 x^{2}$	+	$5 x$
							-	$6 x$	-	$6$
							-	$6 x$	-	$6$

Step 1.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x - 6$

				$x^{2}$	-	$5 x$	-	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$
					+	$5 x^{2}$	+	$5 x$
							-	$6 x$	-	$6$
							+	$6 x$	+	$6$

Step 1.2.5.15

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

				$x^{2}$	-	$5 x$	-	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	-	$11 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	-	$11 x$
					+	$5 x^{2}$	+	$5 x$
							-	$6 x$	-	$6$
							+	$6 x$	+	$6$
										$0$

Step 1.2.5.16

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

$x^{2} - 5 x - 6$

Step 1.2.6

Write $x^{3} - 4 x^{2} - 11 x - 6$ as a set of factors.

$(x + 1) ((x + 1) (x^{2} - 5 x - 6)) = 0$

Step 1.3

Factor $x^{2} - 5 x - 6$ using the AC method.

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Step 1.3.1

Factor $x^{2} - 5 x - 6$ using the AC method.

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Step 1.3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Step 1.3.1.2

Write the factored form using these integers.

$(x + 1) ((x + 1) ((x - 6) (x + 1))) = 0$

Step 1.3.2

Remove unnecessary parentheses.

$(x + 1) ((x + 1) (x - 6) (x + 1)) = 0$

Step 1.4

Combine exponents.

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Step 1.4.1

Combine exponents.

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Step 1.4.1.1

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

$(x + 1) ((x + 1) (x + 1) (x - 6)) = 0$

Step 1.4.1.2

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

$(x + 1) ((x + 1) (x + 1) (x - 6)) = 0$

Step 1.4.1.3

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

$(x + 1) ({(x + 1)}^{1 + 1} (x - 6)) = 0$

Step 1.4.1.4

Add $1$ and $1$ .

$(x + 1) ({(x + 1)}^{2} (x - 6)) = 0$

Step 1.4.2

Remove unnecessary parentheses.

$(x + 1) {(x + 1)}^{2} (x - 6) = 0$

Step 1.5

Combine like factors.

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Step 1.5.1

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

$(x + 1) {(x + 1)}^{2} (x - 6) = 0$

Step 1.5.2

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

${(x + 1)}^{1 + 2} (x - 6) = 0$

Step 1.5.3

Add $1$ and $2$ .

${(x + 1)}^{3} (x - 6) = 0$

Step 2

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

${(x + 1)}^{3} = 0$

$x - 6 = 0$

Step 3

Set ${(x + 1)}^{3}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set ${(x + 1)}^{3}$ equal to $0$ .

${(x + 1)}^{3} = 0$

Step 3.2

Solve ${(x + 1)}^{3} = 0$ for $x$ .

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Step 3.2.1

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 4.2

Add $6$ to both sides of the equation.

$x = 6$

Step 5

The final solution is all the values that make ${(x + 1)}^{3} (x - 6) = 0$ true.

$x = - 1, 6$