Algebra Examples

$x^{4} - x^{3} - 7 x^{2} + x + 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$x^{4} - x^{3} - 7 x^{2} + x + 6 = 0$

Step 1.2

Factor $x^{3}$ out of $x^{4} - x^{3}$ .

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Step 1.2.1

Factor $x^{3}$ out of $x^{4}$ .

$x^{3} x - x^{3} - 7 x^{2} + x + 6 = 0$

Step 1.2.2

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

$x^{3} x + x^{3} \cdot - 1 - 7 x^{2} + x + 6 = 0$

Step 1.2.3

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot - 1$ .

$x^{3} (x - 1) - 7 x^{2} + x + 6 = 0$

Step 1.3

Factor by grouping.

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Step 1.3.1

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 $a \cdot c = - 7 \cdot 6 = - 42$ and whose sum is $b = 1$ .

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Step 1.3.1.1

Multiply by $1$ .

$x^{3} (x - 1) - 7 x^{2} + 1 x + 6 = 0$

Step 1.3.1.2

Rewrite $1$ as $- 6$ plus $7$

$x^{3} (x - 1) - 7 x^{2} + (- 6 + 7) x + 6 = 0$

Step 1.3.1.3

Apply the distributive property.

$x^{3} (x - 1) - 7 x^{2} - 6 x + 7 x + 6 = 0$

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

$x^{3} (x - 1) - 7 x^{2} - 6 x + 7 x + 6 = 0$

Step 1.3.2.2

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

$x^{3} (x - 1) + x (- 7 x - 6) - (- 7 x - 6) = 0$

Step 1.3.3

Factor the polynomial by factoring out the greatest common factor, $- 7 x - 6$ .

$x^{3} (x - 1) + (- 7 x - 6) (x - 1) = 0$

Step 1.4

Factor $x - 1$ out of $x^{3} (x - 1) + (- 7 x - 6) (x - 1)$ .

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Step 1.4.1

Factor $x - 1$ out of $x^{3} (x - 1)$ .

$(x - 1) x^{3} + (- 7 x - 6) (x - 1) = 0$

Step 1.4.2

Factor $x - 1$ out of $(- 7 x - 6) (x - 1)$ .

$(x - 1) x^{3} + (x - 1) (- 7 x - 6) = 0$

Step 1.4.3

Factor $x - 1$ out of $(x - 1) x^{3} + (x - 1) (- 7 x - 6)$ .

$(x - 1) (x^{3} - 7 x - 6) = 0$

Step 1.5

Factor.

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Step 1.5.1

Rewrite $x^{3} - 7 x - 6$ in a factored form.

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Step 1.5.1.1

Factor $x^{3} - 7 x - 6$ using the rational roots test.

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Step 1.5.1.1.1

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 1.5.1.1.2

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 1.5.1.1.3

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 1.5.1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 7 \cdot - 1 - 6$

Step 1.5.1.1.3.2

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

$- 1 - 7 \cdot - 1 - 6$

Step 1.5.1.1.3.3

Multiply $- 7$ by $- 1$ .

$- 1 + 7 - 6$

Step 1.5.1.1.3.4

Add $- 1$ and $7$ .

$6 - 6$

Step 1.5.1.1.3.5

Subtract $6$ from $6$ .

$0$

Step 1.5.1.1.4

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.

$\frac{x^{3} - 7 x - 6}{x + 1}$

Step 1.5.1.1.5

Divide $x^{3} - 7 x - 6$ by $x + 1$ .

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Step 1.5.1.1.5.1

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

$x$

$1$

$x^{3}$

$0 x^{2}$

$7 x$

$6$

Step 1.5.1.1.5.2

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$

Step 1.5.1.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			+	$x^{3}$	+	$x^{2}$

Step 1.5.1.1.5.4

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$

Step 1.5.1.1.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$

Step 1.5.1.1.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$

Step 1.5.1.1.5.7

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$

Step 1.5.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$
					-	$x^{2}$	-	$x$

Step 1.5.1.1.5.9

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$

Step 1.5.1.1.5.10

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$

Step 1.5.1.1.5.11

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$

Step 1.5.1.1.5.12

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

				$x^{2}$	-	$x$	-	$6$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$

Step 1.5.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$6$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$
							-	$6 x$	-	$6$

Step 1.5.1.1.5.14

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

				$x^{2}$	-	$x$	-	$6$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$
							+	$6 x$	+	$6$

Step 1.5.1.1.5.15

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

				$x^{2}$	-	$x$	-	$6$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$7 x$	-	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$
							+	$6 x$	+	$6$
										$0$

Step 1.5.1.1.5.16

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

$x^{2} - x - 6$

Step 1.5.1.1.6

Write $x^{3} - 7 x - 6$ as a set of factors.

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

Step 1.5.1.2

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

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Step 1.5.1.2.1

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

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Step 1.5.1.2.1.1

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 $- 1$ .

$- 3, 2$

Step 1.5.1.2.1.2

Write the factored form using these integers.

$(x - 1) ((x + 1) ((x - 3) (x + 2))) = 0$

Step 1.5.1.2.2

Remove unnecessary parentheses.

$(x - 1) ((x + 1) (x - 3) (x + 2)) = 0$

Step 1.5.2

Remove unnecessary parentheses.

$(x - 1) (x + 1) (x - 3) (x + 2) = 0$

Step 2

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

$x - 1 = 0$

$x + 1 = 0$

$x - 3 = 0$

$x + 2 = 0$

Step 3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

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

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7

The final solution is all the values that make $(x - 1) (x + 1) (x - 3) (x + 2) = 0$ true.

$x = 1, - 1, 3, - 2$