Precalculus Examples

$x^{4} - 625$

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.

$p = \pm 1, \pm 5, \pm 25, \pm 125, \pm 625$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 5, \pm 25, \pm 125, \pm 625$

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.

${(5)}^{4} - 625$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 5$ is a root of the polynomial.

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Step 4.1

Raise $5$ to the power of $4$ .

$625 - 625$

Step 4.2

Subtract $625$ from $625$ .

$0$

Step 5

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 625}{x - 5}$

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.

$5$	$1$	$0$	$0$	$0$	$- 625$

Step 6.2

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

$5$	$1$	$0$	$0$	$0$	$- 625$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(5)$ and place the result of $(5)$ under the next term in the dividend $(0)$ .

$5$	$1$	$0$	$0$	$0$	$- 625$
		$5$
	$1$

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.

$5$	$1$	$0$	$0$	$0$	$- 625$
		$5$
	$1$	$5$

Step 6.5

Multiply the newest entry in the result $(5)$ by the divisor $(5)$ and place the result of $(25)$ under the next term in the dividend $(0)$ .

$5$	$1$	$0$	$0$	$0$	$- 625$
		$5$	$25$
	$1$	$5$

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.

$5$	$1$	$0$	$0$	$0$	$- 625$
		$5$	$25$
	$1$	$5$	$25$

Step 6.7

Multiply the newest entry in the result $(25)$ by the divisor $(5)$ and place the result of $(125)$ under the next term in the dividend $(0)$ .

$5$	$1$	$0$	$0$	$0$	$- 625$
		$5$	$25$	$125$
	$1$	$5$	$25$

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.

$5$	$1$	$0$	$0$	$0$	$- 625$
		$5$	$25$	$125$
	$1$	$5$	$25$	$125$

Step 6.9

Multiply the newest entry in the result $(125)$ by the divisor $(5)$ and place the result of $(625)$ under the next term in the dividend $(- 625)$ .

$5$	$1$	$0$	$0$	$0$	$- 625$
		$5$	$25$	$125$	$625$
	$1$	$5$	$25$	$125$

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.

$5$	$1$	$0$	$0$	$0$	$- 625$
		$5$	$25$	$125$	$625$
	$1$	$5$	$25$	$125$	$0$

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.

$1 x^{3} + 5 x^{2} + (25) x + 125$

Step 6.12

Simplify the quotient polynomial.

$x^{3} + 5 x^{2} + 25 x + 125$

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.

$(x - 5) (x^{3} + 5 x^{2}) + 25 x + 125$

Step 7.2

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

$(x - 5) + x^{2} (x + 5) + 25 (x + 5)$

$(x - 5) x^{2} (x + 5) + 25 (x + 5)$

Step 8

Factor the polynomial by factoring out the greatest common factor, $x + 5$ .

$(x - 5) (x + 5) (x^{2} + 25)$

Step 9

Add $625$ to both sides of the equation.

$x^{4} = 625$

Step 10

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

$x = \pm \sqrt[4]{625}$

Step 11

Simplify $\pm \sqrt[4]{625}$ .

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Step 11.1

Rewrite $625$ as $5^{4}$ .

$x = \pm \sqrt[4]{5^{4}}$

Step 11.2

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

$x = \pm 5$

Step 12

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

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Step 12.1

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

$x = 5$

Step 12.2

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

$x = - 5$

Step 12.3

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

$x = 5, - 5$

Step 13