Finite Math Examples

$4 x^{4} + 15 x^{2} - 4$

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.

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

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

Step 2

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

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

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.

$4 {(\frac{1}{2})}^{4} + 15 {(\frac{1}{2})}^{2} - 4$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = \frac{1}{2}$ is a root of the polynomial.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Apply the product rule to $\frac{1}{2}$ .

$4 \frac{1^{4}}{2^{4}} + 15 {(\frac{1}{2})}^{2} - 4$

Step 4.1.2

One to any power is one.

$4 \frac{1}{2^{4}} + 15 {(\frac{1}{2})}^{2} - 4$

Step 4.1.3

Raise $2$ to the power of $4$ .

$4 (\frac{1}{16}) + 15 {(\frac{1}{2})}^{2} - 4$

Step 4.1.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.1.4.1

Factor $4$ out of $16$ .

$4 \frac{1}{4 (4)} + 15 {(\frac{1}{2})}^{2} - 4$

Step 4.1.4.2

Cancel the common factor.

$4 \frac{1}{4 \cdot 4} + 15 {(\frac{1}{2})}^{2} - 4$

Step 4.1.4.3

Rewrite the expression.

$\frac{1}{4} + 15 {(\frac{1}{2})}^{2} - 4$

Step 4.1.5

Apply the product rule to $\frac{1}{2}$ .

$\frac{1}{4} + 15 \frac{1^{2}}{2^{2}} - 4$

Step 4.1.6

One to any power is one.

$\frac{1}{4} + 15 \frac{1}{2^{2}} - 4$

Step 4.1.7

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

$\frac{1}{4} + 15 (\frac{1}{4}) - 4$

Step 4.1.8

Combine $15$ and $\frac{1}{4}$ .

$\frac{1}{4} + \frac{15}{4} - 4$

Step 4.2

Combine fractions.

Tap for more steps...

Step 4.2.1

Combine the numerators over the common denominator.

$- 4 + \frac{1 + 15}{4}$

Step 4.2.2

Simplify the expression.

Tap for more steps...

Step 4.2.2.1

Add $1$ and $15$ .

$- 4 + \frac{16}{4}$

Step 4.2.2.2

Divide $16$ by $4$ .

$- 4 + 4$

Step 4.2.2.3

Add $- 4$ and $4$ .

$0$

Step 5

Since $\frac{1}{2}$ is a known root, divide the polynomial by $x - \frac{1}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{4} + 15 x^{2} - 4}{x - \frac{1}{2}}$

Step 6

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

Tap for more steps...

Step 6.1

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

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$

Step 6.2

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

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$

	$4$

Step 6.3

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{1}{2})$ and place the result of $(2)$ under the next term in the dividend $(0)$ .

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$
		$2$
	$4$

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.

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$
		$2$
	$4$	$2$

Step 6.5

Multiply the newest entry in the result $(2)$ by the divisor $(\frac{1}{2})$ and place the result of $(1)$ under the next term in the dividend $(15)$ .

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$
		$2$	$1$
	$4$	$2$

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.

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$
		$2$	$1$
	$4$	$2$	$16$

Step 6.7

Multiply the newest entry in the result $(16)$ by the divisor $(\frac{1}{2})$ and place the result of $(8)$ under the next term in the dividend $(0)$ .

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$
		$2$	$1$	$8$
	$4$	$2$	$16$

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.

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$
		$2$	$1$	$8$
	$4$	$2$	$16$	$8$

Step 6.9

Multiply the newest entry in the result $(8)$ by the divisor $(\frac{1}{2})$ and place the result of $(4)$ under the next term in the dividend $(- 4)$ .

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$
		$2$	$1$	$8$	$4$
	$4$	$2$	$16$	$8$

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.

$\frac{1}{2}$	$4$	$0$	$15$	$0$	$- 4$
		$2$	$1$	$8$	$4$
	$4$	$2$	$16$	$8$	$0$

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.

$4 x^{3} + 2 x^{2} + (16) x + 8$

Step 6.12

Simplify the quotient polynomial.

$4 x^{3} + 2 x^{2} + 16 x + 8$

Step 7

Factor $2$ out of $4 x^{3} + 2 x^{2} + 16 x + 8$ .

Tap for more steps...

Step 7.1

Factor $2$ out of $4 x^{3}$ .

$(x - \frac{1}{2}) (2 (2 x^{3}) + 2 x^{2} + 16 x + 8)$

Step 7.2

Factor $2$ out of $2 x^{2}$ .

$(x - \frac{1}{2}) (2 (2 x^{3}) + 2 (x^{2}) + 16 x + 8)$

Step 7.3

Factor $2$ out of $16 x$ .

$(x - \frac{1}{2}) (2 (2 x^{3}) + 2 (x^{2}) + 2 (8 x) + 8)$

Step 7.4

Factor $2$ out of $8$ .

$(x - \frac{1}{2}) (2 (2 x^{3}) + 2 x^{2} + 2 (8 x) + 2 \cdot 4)$

Step 7.5

Factor $2$ out of $2 (2 x^{3}) + 2 x^{2}$ .

$(x - \frac{1}{2}) (2 (2 x^{3} + x^{2}) + 2 (8 x) + 2 \cdot 4)$

Step 7.6

Factor $2$ out of $2 (2 x^{3} + x^{2}) + 2 (8 x)$ .

$(x - \frac{1}{2}) (2 (2 x^{3} + x^{2} + 8 x) + 2 \cdot 4)$

Step 7.7

Factor $2$ out of $2 (2 x^{3} + x^{2} + 8 x) + 2 \cdot 4$ .

$(x - \frac{1}{2}) (2 (2 x^{3} + x^{2} + 8 x + 4))$

Step 8

Factor out the greatest common factor from each group.

Tap for more steps...

Step 8.1

Group the first two terms and the last two terms.

$(x - \frac{1}{2}) (2 ((2 x^{3} + x^{2}) + 8 x + 4))$

Step 8.2

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

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

Step 9

Factor.

Tap for more steps...

Step 9.1

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Step 9.2

Remove unnecessary parentheses.

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

Step 10

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

$4 u^{2} + 15 u - 4 = 0$

$u = x^{2}$

Step 11

Factor by grouping.

Tap for more steps...

Step 11.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 4 = - 16$ and whose sum is $b = 15$ .

Tap for more steps...

Step 11.1.1

Factor $15$ out of $15 u$ .

$4 u^{2} + 15 (u) - 4 = 0$

Step 11.1.2

Rewrite $15$ as $- 1$ plus $16$

$4 u^{2} + (- 1 + 16) u - 4 = 0$

Step 11.1.3

Apply the distributive property.

$4 u^{2} - 1 u + 16 u - 4 = 0$

Step 11.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 11.2.1

Group the first two terms and the last two terms.

$(4 u^{2} - 1 u) + 16 u - 4 = 0$

Step 11.2.2

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

$u (4 u - 1) + 4 (4 u - 1) = 0$

Step 11.3

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$(4 u - 1) (u + 4) = 0$

Step 12

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

$4 u - 1 = 0$

$u + 4 = 0$

Step 13

Set $4 u - 1$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 13.1

Set $4 u - 1$ equal to $0$ .

$4 u - 1 = 0$

Step 13.2

Solve $4 u - 1 = 0$ for $u$ .

Tap for more steps...

Step 13.2.1

Add $1$ to both sides of the equation.

$4 u = 1$

Step 13.2.2

Divide each term in $4 u = 1$ by $4$ and simplify.

Tap for more steps...

Step 13.2.2.1

Divide each term in $4 u = 1$ by $4$ .

$\frac{4 u}{4} = \frac{1}{4}$

Step 13.2.2.2

Simplify the left side.

Tap for more steps...

Step 13.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 13.2.2.2.1.1

Cancel the common factor.

$\frac{4 u}{4} = \frac{1}{4}$

Step 13.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{4}$

Step 14

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

Tap for more steps...

Step 14.1

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

$u + 4 = 0$

Step 14.2

Subtract $4$ from both sides of the equation.

$u = - 4$

Step 15

The final solution is all the values that make $(4 u - 1) (u + 4) = 0$ true.

$u = \frac{1}{4}, - 4$

Step 16

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

$x^{2} = \frac{1}{4}$

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

Step 17

Solve the first equation for $x$ .

$x^{2} = \frac{1}{4}$

Step 18

Solve the equation for $x$ .

Tap for more steps...

Step 18.1

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

$x = \pm \sqrt{\frac{1}{4}}$

Step 18.2

Simplify $\pm \sqrt{\frac{1}{4}}$ .

Tap for more steps...

Step 18.2.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 18.2.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{4}}$

Step 18.2.3

Simplify the denominator.

Tap for more steps...

Step 18.2.3.1

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

$x = \pm \frac{1}{\sqrt{2^{2}}}$

Step 18.2.3.2

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

$x = \pm \frac{1}{2}$

Step 18.3

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

Tap for more steps...

Step 18.3.1

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

$x = \frac{1}{2}$

Step 18.3.2

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

$x = - \frac{1}{2}$

Step 18.3.3

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

$x = \frac{1}{2}, - \frac{1}{2}$

Step 19

Solve the second equation for $x$ .

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

Step 20

Solve the equation for $x$ .

Tap for more steps...

Step 20.1

Remove parentheses.

$x^{2} = - 4$

Step 20.2

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

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

Step 20.3

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

Tap for more steps...

Step 20.3.1

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

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

Step 20.3.2

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

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

Step 20.3.3

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

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

Step 20.3.4

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

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

Step 20.3.5

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

$x = \pm i \cdot 2$

Step 20.3.6

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

$x = \pm 2 i$

Step 20.4

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

Tap for more steps...

Step 20.4.1

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

$x = 2 i$

Step 20.4.2

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

$x = - 2 i$

Step 20.4.3

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

$x = 2 i, - 2 i$

Step 21

The solution to $4 x^{4} + 15 x^{2} - 4 = 0$ is $x = \frac{1}{2}, - \frac{1}{2}, 2 i, - 2 i$ .

$x = \frac{1}{2}, - \frac{1}{2}, 2 i, - 2 i$

Step 22