Algebra Examples

$f (x) = 4 x^{2} - 25$

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.

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

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

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}, \pm 25, \pm \frac{25}{2}, \pm \frac{25}{4}$

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.

$4 {(\frac{5}{2})}^{2} - 25$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = \frac{5}{2}$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the product rule to $\frac{5}{2}$ .

$4 \frac{5^{2}}{2^{2}} - 25$

Step 4.1.2

Raise $5$ to the power of $2$ .

$4 \frac{25}{2^{2}} - 25$

Step 4.1.3

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

$4 (\frac{25}{4}) - 25$

Step 4.1.4

Cancel the common factor of $4$ .

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Step 4.1.4.1

Cancel the common factor.

$4 (\frac{25}{4}) - 25$

Step 4.1.4.2

Rewrite the expression.

$25 - 25$

Step 4.2

Subtract $25$ from $25$ .

$0$

Step 5

Since $\frac{5}{2}$ is a known root, divide the polynomial by $x - \frac{5}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{2} - 25}{x - \frac{5}{2}}$

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.

$\frac{5}{2}$	$4$	$0$	$- 25$

Step 6.2

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$\frac{5}{2}$	$4$	$0$	$- 25$

	$4$

Step 6.3

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{5}{2})$ and place the result of $(10)$ under the next term in the dividend $(0)$ .

$\frac{5}{2}$	$4$	$0$	$- 25$
		$10$
	$4$

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.

$\frac{5}{2}$	$4$	$0$	$- 25$
		$10$
	$4$	$10$

Step 6.5

Multiply the newest entry in the result $(10)$ by the divisor $(\frac{5}{2})$ and place the result of $(25)$ under the next term in the dividend $(- 25)$ .

$\frac{5}{2}$	$4$	$0$	$- 25$
		$10$	$25$
	$4$	$10$

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.

$\frac{5}{2}$	$4$	$0$	$- 25$
		$10$	$25$
	$4$	$10$	$0$

Step 6.7

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

$(4) x + 10$

Step 6.8

Simplify the quotient polynomial.

$4 x + 10$

Step 7

Factor $2$ out of $4 x + 10$ .

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Step 7.1

Factor $2$ out of $4 x$ .

$(x - \frac{5}{2}) (2 (2 x) + 10)$

Step 7.2

Factor $2$ out of $10$ .

$(x - \frac{5}{2}) (2 (2 x) + 2 (5))$

Step 7.3

Factor $2$ out of $2 (2 x) + 2 (5)$ .

$(x - \frac{5}{2}) (2 (2 x + 5))$

Step 8

Add $25$ to both sides of the equation.

$4 x^{2} = 25$

Step 9

Divide each term in $4 x^{2} = 25$ by $4$ and simplify.

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Step 9.1

Divide each term in $4 x^{2} = 25$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{25}{4}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $4$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{25}{4}$

Step 9.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{25}{4}$

Step 10

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

$x = \pm \sqrt{\frac{25}{4}}$

Step 11

Simplify $\pm \sqrt{\frac{25}{4}}$ .

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Step 11.1

Rewrite $\sqrt{\frac{25}{4}}$ as $\frac{\sqrt{25}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{25}}{\sqrt{4}}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \frac{\sqrt{5^{2}}}{\sqrt{4}}$

Step 11.2.2

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

$x = \pm \frac{5}{\sqrt{4}}$

Step 11.3

Simplify the denominator.

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Step 11.3.1

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

$x = \pm \frac{5}{\sqrt{2^{2}}}$

Step 11.3.2

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

$x = \pm \frac{5}{2}$

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.

$x = \frac{5}{2}$

Step 12.2

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

$x = - \frac{5}{2}$

Step 12.3

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

$x = \frac{5}{2}, - \frac{5}{2}$

Step 13