Precalculus Examples

$x^{6} - 9 x^{4} + 24 x^{2} - 16 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$x^{6} - 9 x^{4} + 24 x^{2} - 16 = 0$

Step 1.2

Factor $- 1$ out of $- 9 x^{4} + 24 x^{2} - 16$ .

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Step 1.2.1

Factor $- 1$ out of $- 9 x^{4}$ .

$x^{6} - (9 x^{4}) + 24 x^{2} - 16 = 0$

Step 1.2.2

Factor $- 1$ out of $24 x^{2}$ .

$x^{6} - (9 x^{4}) - (- 24 x^{2}) - 16 = 0$

Step 1.2.3

Rewrite $- 16$ as $- 1 (16)$ .

$x^{6} - (9 x^{4}) - (- 24 x^{2}) - 1 \cdot 16 = 0$

Step 1.2.4

Factor $- 1$ out of $- (9 x^{4}) - (- 24 x^{2})$ .

$x^{6} - (9 x^{4} - 24 x^{2}) - 1 \cdot 16 = 0$

Step 1.2.5

Factor $- 1$ out of $- (9 x^{4} - 24 x^{2}) - 1 (16)$ .

$x^{6} - (9 x^{4} - 24 x^{2} + 16) = 0$

Step 1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x^{6} - (9 {(x^{2})}^{2} - 24 x^{2} + 16) = 0$

Step 1.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x^{6} - (9 u^{2} - 24 u + 16) = 0$

Step 1.5

Factor using the perfect square rule.

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Step 1.5.1

Rewrite $9 u^{2}$ as ${(3 u)}^{2}$ .

$x^{6} - ({(3 u)}^{2} - 24 u + 16) = 0$

Step 1.5.2

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

$x^{6} - ({(3 u)}^{2} - 24 u + 4^{2}) = 0$

Step 1.5.3

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

$24 u = 2 \cdot (3 u) \cdot 4$

Step 1.5.4

Rewrite the polynomial.

$x^{6} - ({(3 u)}^{2} - 2 \cdot (3 u) \cdot 4 + 4^{2}) = 0$

Step 1.5.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 u$ and $b = 4$ .

$x^{6} - {(3 u - 4)}^{2} = 0$

Step 1.6

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 1.7

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

${(x^{3})}^{2} - {(3 x^{2} - 4)}^{2} = 0$

Step 1.8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{3}$ and $b = 3 x^{2} - 4$ .

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

Step 1.9

Simplify.

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Step 1.9.1

Rewrite $x^{3} + 3 x^{2} - 4$ in a factored form.

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Step 1.9.1.1

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

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Step 1.9.1.1.1

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.

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

$q = \pm 1$

Step 1.9.1.1.2

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

$\pm 1, \pm 4, \pm 2$

Step 1.9.1.1.3

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 1.9.1.1.3.1

Substitute $1$ into the polynomial.

$1^{3} + 3 \cdot 1^{2} - 4$

Step 1.9.1.1.3.2

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

$1 + 3 \cdot 1^{2} - 4$

Step 1.9.1.1.3.3

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

$1 + 3 \cdot 1 - 4$

Step 1.9.1.1.3.4

Multiply $3$ by $1$ .

$1 + 3 - 4$

Step 1.9.1.1.3.5

Add $1$ and $3$ .

$4 - 4$

Step 1.9.1.1.3.6

Subtract $4$ from $4$ .

$0$

Step 1.9.1.1.4

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.

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

Step 1.9.1.1.5

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

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Step 1.9.1.1.5.1

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

$x$

$1$

$x^{3}$

$3 x^{2}$

$0 x$

$4$

Step 1.9.1.1.5.2

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

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

Step 1.9.1.1.5.3

Multiply the new quotient term by the divisor.

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

Step 1.9.1.1.5.4

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

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

Step 1.9.1.1.5.5

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

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

Step 1.9.1.1.5.6

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

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

Step 1.9.1.1.5.7

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

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

Step 1.9.1.1.5.8

Multiply the new quotient term by the divisor.

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

Step 1.9.1.1.5.9

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

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

Step 1.9.1.1.5.10

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

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

Step 1.9.1.1.5.11

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

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

Step 1.9.1.1.5.12

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

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

Step 1.9.1.1.5.13

Multiply the new quotient term by the divisor.

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

Step 1.9.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x - 4$

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

Step 1.9.1.1.5.15

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

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

Step 1.9.1.1.5.16

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

$x^{2} + 4 x + 4$

Step 1.9.1.1.6

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

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

Step 1.9.1.2

Factor using the perfect square rule.

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Step 1.9.1.2.1

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

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

Step 1.9.1.2.2

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

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

Step 1.9.1.2.3

Rewrite the polynomial.

$(x - 1) (x^{2} + 2 \cdot x \cdot 2 + 2^{2}) (x^{3} - (3 x^{2} - 4)) = 0$

Step 1.9.1.2.4

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

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

Step 1.9.2

Apply the distributive property.

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

Step 1.9.3

Multiply $3$ by $- 1$ .

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

Step 1.9.4

Multiply $- 1$ by $- 4$ .

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

Step 1.9.5

Factor.

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Step 1.9.5.1

Rewrite $x^{3} - 3 x^{2} + 4$ in a factored form.

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Step 1.9.5.1.1

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

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Step 1.9.5.1.1.1

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.

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

$q = \pm 1$

Step 1.9.5.1.1.2

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

$\pm 1, \pm 4, \pm 2$

Step 1.9.5.1.1.3

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 1.9.5.1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 3 {(- 1)}^{2} + 4$

Step 1.9.5.1.1.3.2

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

$- 1 - 3 {(- 1)}^{2} + 4$

Step 1.9.5.1.1.3.3

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

$- 1 - 3 \cdot 1 + 4$

Step 1.9.5.1.1.3.4

Multiply $- 3$ by $1$ .

$- 1 - 3 + 4$

Step 1.9.5.1.1.3.5

Subtract $3$ from $- 1$ .

$- 4 + 4$

Step 1.9.5.1.1.3.6

Add $- 4$ and $4$ .

$0$

Step 1.9.5.1.1.4

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.

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

Step 1.9.5.1.1.5

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

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Step 1.9.5.1.1.5.1

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

$x$

$1$

$x^{3}$

$3 x^{2}$

$0 x$

$4$

Step 1.9.5.1.1.5.2

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

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

Step 1.9.5.1.1.5.3

Multiply the new quotient term by the divisor.

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

Step 1.9.5.1.1.5.4

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

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

Step 1.9.5.1.1.5.5

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

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

Step 1.9.5.1.1.5.6

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

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

Step 1.9.5.1.1.5.7

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

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

Step 1.9.5.1.1.5.8

Multiply the new quotient term by the divisor.

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

Step 1.9.5.1.1.5.9

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

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

Step 1.9.5.1.1.5.10

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

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

Step 1.9.5.1.1.5.11

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

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

Step 1.9.5.1.1.5.12

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

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

Step 1.9.5.1.1.5.13

Multiply the new quotient term by the divisor.

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

Step 1.9.5.1.1.5.14

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

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

Step 1.9.5.1.1.5.15

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

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

Step 1.9.5.1.1.5.16

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

$x^{2} - 4 x + 4$

Step 1.9.5.1.1.6

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

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

Step 1.9.5.1.2

Factor using the perfect square rule.

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Step 1.9.5.1.2.1

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

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

Step 1.9.5.1.2.2

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

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

Step 1.9.5.1.2.3

Rewrite the polynomial.

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

Step 1.9.5.1.2.4

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

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

Step 1.9.5.2

Remove unnecessary parentheses.

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

Step 2

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

$x - 1 = 0$

${(x + 2)}^{2} = 0$

$x + 1 = 0$

${(x - 2)}^{2} = 0$

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 ${(x + 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set ${(x + 2)}^{2}$ equal to $0$ .

${(x + 2)}^{2} = 0$

Step 4.2

Solve ${(x + 2)}^{2} = 0$ for $x$ .

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Step 4.2.1

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

$x + 2 = 0$

Step 4.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

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

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Step 5.1

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

$x + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 6.1

Set ${(x - 2)}^{2}$ equal to $0$ .

${(x - 2)}^{2} = 0$

Step 6.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 6.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 6.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 7

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

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

Step 8