Algebra Examples

$x^{4} - 13 x^{2} + 36$

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.

${(2)}^{4} - 13 {(2)}^{2} + 36$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 2$ 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 $2$ to the power of $4$ .

$16 - 13 {(2)}^{2} + 36$

Step 4.1.2

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

$16 - 13 \cdot 4 + 36$

Step 4.1.3

Multiply $- 13$ by $4$ .

$16 - 52 + 36$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $52$ from $16$ .

$- 36 + 36$

Step 4.2.2

Add $- 36$ and $36$ .

$0$

Step 5

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 13 x^{2} + 36}{x - 2}$

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.

$2$	$1$	$0$	$- 13$	$0$	$36$

Step 6.2

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

$2$	$1$	$0$	$- 13$	$0$	$36$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(2)$ and place the result of $(2)$ under the next term in the dividend $(0)$ .

$2$	$1$	$0$	$- 13$	$0$	$36$
		$2$
	$1$

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.

$2$	$1$	$0$	$- 13$	$0$	$36$
		$2$
	$1$	$2$

Step 6.5

Multiply the newest entry in the result $(2)$ by the divisor $(2)$ and place the result of $(4)$ under the next term in the dividend $(- 13)$ .

$2$	$1$	$0$	$- 13$	$0$	$36$
		$2$	$4$
	$1$	$2$

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.

$2$	$1$	$0$	$- 13$	$0$	$36$
		$2$	$4$
	$1$	$2$	$- 9$

Step 6.7

Multiply the newest entry in the result $(- 9)$ by the divisor $(2)$ and place the result of $(- 18)$ under the next term in the dividend $(0)$ .

$2$	$1$	$0$	$- 13$	$0$	$36$
		$2$	$4$	$- 18$
	$1$	$2$	$- 9$

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.

$2$	$1$	$0$	$- 13$	$0$	$36$
		$2$	$4$	$- 18$
	$1$	$2$	$- 9$	$- 18$

Step 6.9

Multiply the newest entry in the result $(- 18)$ by the divisor $(2)$ and place the result of $(- 36)$ under the next term in the dividend $(36)$ .

$2$	$1$	$0$	$- 13$	$0$	$36$
		$2$	$4$	$- 18$	$- 36$
	$1$	$2$	$- 9$	$- 18$

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.

$2$	$1$	$0$	$- 13$	$0$	$36$
		$2$	$4$	$- 18$	$- 36$
	$1$	$2$	$- 9$	$- 18$	$0$

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.

$1 x^{3} + 2 x^{2} + (- 9) x - 18$

Step 6.12

Simplify the quotient polynomial.

$x^{3} + 2 x^{2} - 9 x - 18$

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.

$(x - 2) (x^{3} + 2 x^{2}) - 9 x - 18$

Step 7.2

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

$(x - 2) + x^{2} (x + 2) - 9 (x + 2)$

$(x - 2) x^{2} (x + 2) - 9 (x + 2)$

Step 8

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$(x - 2) (x + 2) (x^{2} - 9)$

Step 9

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

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

Step 10

Factor.

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Step 10.1

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

$(x - 2) + (x + 2) ((x + 3) (x - 3))$

Step 10.2

Remove unnecessary parentheses.

$(x - 2) + (x + 2) (x + 3) (x - 3)$

$(x - 2) (x + 2) (x + 3) (x - 3)$

Step 11

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

$u^{2} - 13 u + 36 = 0$

$u = x^{2}$

Step 12

Factor $u^{2} - 13 u + 36$ using the AC method.

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Step 12.1

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 $- 13$ .

$- 9, - 4$

Step 12.2

Write the factored form using these integers.

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

Step 13

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 14

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

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Step 14.1

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

$u - 9 = 0$

Step 14.2

Add $9$ to both sides of the equation.

$u = 9$

Step 15

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

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Step 15.1

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

$u - 4 = 0$

Step 15.2

Add $4$ to both sides of the equation.

$u = 4$

Step 16

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

$u = 9, 4$

Step 17

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

$x^{2} = 9$

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

Step 18

Solve the first equation for $x$ .

$x^{2} = 9$

Step 19

Solve the equation for $x$ .

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Step 19.1

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

$x = \pm \sqrt{9}$

Step 19.2

Simplify $\pm \sqrt{9}$ .

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Step 19.2.1

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

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

Step 19.2.2

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

$x = \pm 3$

Step 19.3

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

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Step 19.3.1

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

$x = 3$

Step 19.3.2

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

$x = - 3$

Step 19.3.3

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

$x = 3, - 3$

Step 20

Solve the second equation for $x$ .

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

Step 21

Solve the equation for $x$ .

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Step 21.1

Remove parentheses.

$x^{2} = 4$

Step 21.2

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

$x = \pm \sqrt{4}$

Step 21.3

Simplify $\pm \sqrt{4}$ .

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Step 21.3.1

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

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

Step 21.3.2

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

$x = \pm 2$

Step 21.4

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

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Step 21.4.1

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

$x = 2$

Step 21.4.2

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

$x = - 2$

Step 21.4.3

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

$x = 2, - 2$

Step 22

The solution to $x^{4} - 13 x^{2} + 36 = 0$ is $x = 3, - 3, 2, - 2$ .

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

Step 23