Precalculus Examples

$x^{4} - 5 x^{2} - 36$

Step 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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(3)}^{4} - 5 {(3)}^{2} - 36$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 3$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $3$ to the power of $4$ .

$81 - 5 {(3)}^{2} - 36$

Step 4.1.2

Raise $3$ to the power of $2$ .

$81 - 5 \cdot 9 - 36$

Step 4.1.3

Multiply $- 5$ by $9$ .

$81 - 45 - 36$

Step 4.2

Simplify by subtracting numbers.

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Step 4.2.1

Subtract $45$ from $81$ .

$36 - 36$

Step 4.2.2

Subtract $36$ from $36$ .

$0$

Step 5

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 5 x^{2} - 36}{x - 3}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$3$	$1$	$0$	$- 5$	$0$	$- 36$

Step 6.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$3$	$1$	$0$	$- 5$	$0$	$- 36$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(3)$ and place the result of $(3)$ under the next term in the dividend $(0)$ .

$3$	$1$	$0$	$- 5$	$0$	$- 36$
		$3$
	$1$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 5$	$0$	$- 36$
		$3$
	$1$	$3$

Step 6.5

Multiply the newest entry in the result $(3)$ by the divisor $(3)$ and place the result of $(9)$ under the next term in the dividend $(- 5)$ .

$3$	$1$	$0$	$- 5$	$0$	$- 36$
		$3$	$9$
	$1$	$3$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 5$	$0$	$- 36$
		$3$	$9$
	$1$	$3$	$4$

Step 6.7

Multiply the newest entry in the result $(4)$ by the divisor $(3)$ and place the result of $(12)$ under the next term in the dividend $(0)$ .

$3$	$1$	$0$	$- 5$	$0$	$- 36$
		$3$	$9$	$12$
	$1$	$3$	$4$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 5$	$0$	$- 36$
		$3$	$9$	$12$
	$1$	$3$	$4$	$12$

Step 6.9

Multiply the newest entry in the result $(12)$ by the divisor $(3)$ and place the result of $(36)$ under the next term in the dividend $(- 36)$ .

$3$	$1$	$0$	$- 5$	$0$	$- 36$
		$3$	$9$	$12$	$36$
	$1$	$3$	$4$	$12$

Step 6.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$3$	$1$	$0$	$- 5$	$0$	$- 36$
		$3$	$9$	$12$	$36$
	$1$	$3$	$4$	$12$	$0$

Step 6.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 3 x^{2} + (4) x + 12$

Step 6.12

Simplify the quotient polynomial.

$x^{3} + 3 x^{2} + 4 x + 12$

Step 7

Factor out the greatest common factor from each group.

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Step 7.1

Group the first two terms and the last two terms.

$(x - 3) (x^{3} + 3 x^{2}) + 4 x + 12$

Step 7.2

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

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

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

Step 8

Factor the polynomial by factoring out the greatest common factor, $x + 3$ .

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

Step 9

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 5 u - 36 = 0$

$u = x^{2}$

Step 10

Factor $u^{2} - 5 u - 36$ using the AC method.

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Step 10.1

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 $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Step 10.2

Write the factored form using these integers.

$(u - 9) (u + 4) = 0$

Step 11

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

$u - 9 = 0$

$u + 4 = 0$

Step 12

Set $u - 9$ equal to $0$ and solve for $u$ .

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Step 12.1

Set $u - 9$ equal to $0$ .

$u - 9 = 0$

Step 12.2

Add $9$ to both sides of the equation.

$u = 9$

Step 13

Set $u + 4$ equal to $0$ and solve for $u$ .

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Step 13.1

Set $u + 4$ equal to $0$ .

$u + 4 = 0$

Step 13.2

Subtract $4$ from both sides of the equation.

$u = - 4$

Step 14

The final solution is all the values that make $(u - 9) (u + 4) = 0$ true.

$u = 9, - 4$

Step 15

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 9$

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

Step 16

Solve the first equation for $x$ .

$x^{2} = 9$

Step 17

Solve the equation for $x$ .

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Step 17.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

Step 17.2

Simplify $\pm \sqrt{9}$ .

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Step 17.2.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 17.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 17.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 17.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 17.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

Step 17.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

Step 18

Solve the second equation for $x$ .

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

Step 19

Solve the equation for $x$ .

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Step 19.1

Remove parentheses.

$x^{2} = - 4$

Step 19.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 4}$

Step 19.3

Simplify $\pm \sqrt{- 4}$ .

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Step 19.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 19.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 19.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{4}$

Step 19.3.4

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

$x = \pm i \cdot \sqrt{2^{2}}$

Step 19.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 2$

Step 19.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 19.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 19.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 i$

Step 19.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i$

Step 19.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i, - 2 i$

Step 20

The solution to $x^{4} - 5 x^{2} - 36 = 0$ is $x = 3, - 3, 2 i, - 2 i$ .

$x = 3, - 3, 2 i, - 2 i$

Step 21