Finite Math Examples

$4 x^{3} - 2 x^{2} + 48 x$

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.

$p = \pm 0$

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

Step 2

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

$0$

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.

$4 {(0)}^{3} - 2 {(0)}^{2} + 48 (0)$

Step 4

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

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Step 4.1

Simplify each term.

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Step 4.1.1

Raising $0$ to any positive power yields $0$ .

$4 \cdot 0 - 2 {(0)}^{2} + 48 (0)$

Step 4.1.2

Multiply $4$ by $0$ .

$0 - 2 {(0)}^{2} + 48 (0)$

Step 4.1.3

Raising $0$ to any positive power yields $0$ .

$0 - 2 \cdot 0 + 48 (0)$

Step 4.1.4

Multiply $- 2$ by $0$ .

$0 + 0 + 48 (0)$

Step 4.1.5

Multiply $48$ by $0$ .

$0 + 0 + 0$

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add $0$ and $0$ .

$0 + 0$

Step 4.2.2

Add $0$ and $0$ .

$0$

Step 5

Since $0$ is a known root, divide the polynomial by $x + 0$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 2 x^{2} + 48 x}{x + 0}$

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.

$0$	$4$	$- 2$	$48$	$0$

Step 6.2

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

$0$	$4$	$- 2$	$48$	$0$

	$4$

Step 6.3

Multiply the newest entry in the result $(4)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 2)$ .

$0$	$4$	$- 2$	$48$	$0$
		$0$
	$4$

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.

$0$	$4$	$- 2$	$48$	$0$
		$0$
	$4$	$- 2$

Step 6.5

Multiply the newest entry in the result $(- 2)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(48)$ .

$0$	$4$	$- 2$	$48$	$0$
		$0$	$0$
	$4$	$- 2$

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.

$0$	$4$	$- 2$	$48$	$0$
		$0$	$0$
	$4$	$- 2$	$48$

Step 6.7

Multiply the newest entry in the result $(48)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$4$	$- 2$	$48$	$0$
		$0$	$0$	$0$
	$4$	$- 2$	$48$

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.

$0$	$4$	$- 2$	$48$	$0$
		$0$	$0$	$0$
	$4$	$- 2$	$48$	$0$

Step 6.9

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

$4 x^{2} + - 2 x + 48$

Step 6.10

Simplify the quotient polynomial.

$4 x^{2} - 2 x + 48$

Step 7

Factor $2$ out of $4 x^{2} - 2 x + 48$ .

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Step 7.1

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

$(x + 0) (2 (2 x^{2}) - 2 x + 48)$

Step 7.2

Factor $2$ out of $- 2 x$ .

$(x + 0) (2 (2 x^{2}) + 2 (- x) + 48)$

Step 7.3

Factor $2$ out of $48$ .

$(x + 0) (2 (2 x^{2}) + 2 (- x) + 2 (24))$

Step 7.4

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

$(x + 0) (2 (2 x^{2} - x) + 2 (24))$

Step 7.5

Factor $2$ out of $2 (2 x^{2} - x) + 2 (24)$ .

$(x + 0) (2 (2 x^{2} - x + 24))$

Step 8

Factor $2 x$ out of $4 x^{3} - 2 x^{2} + 48 x$ .

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Step 8.1

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

$2 x (2 x^{2}) - 2 x^{2} + 48 x = 0$

Step 8.2

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

$2 x (2 x^{2}) + 2 x (- x) + 48 x = 0$

Step 8.3

Factor $2 x$ out of $48 x$ .

$2 x (2 x^{2}) + 2 x (- x) + 2 x (24) = 0$

Step 8.4

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

$2 x (2 x^{2} - x) + 2 x (24) = 0$

Step 8.5

Factor $2 x$ out of $2 x (2 x^{2} - x) + 2 x (24)$ .

$2 x (2 x^{2} - x + 24) = 0$

Step 9

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

$x = 0$

$2 x^{2} - x + 24 = 0$

Step 10

Set $x$ equal to $0$ .

$x = 0$

Step 11

Set $2 x^{2} - x + 24$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $2 x^{2} - x + 24$ equal to $0$ .

$2 x^{2} - x + 24 = 0$

Step 11.2

Solve $2 x^{2} - x + 24 = 0$ for $x$ .

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Step 11.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 11.2.2

Substitute the values $a = 2$ , $b = - 1$ , and $c = 24$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (2 \cdot 24)}}{2 \cdot 2}$

Step 11.2.3

Simplify.

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Step 11.2.3.1

Simplify the numerator.

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Step 11.2.3.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot 24}}{2 \cdot 2}$

Step 11.2.3.1.2

Multiply $- 4 \cdot 2 \cdot 24$ .

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Step 11.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot 24}}{2 \cdot 2}$

Step 11.2.3.1.2.2

Multiply $- 8$ by $24$ .

$x = \frac{1 \pm \sqrt{1 - 192}}{2 \cdot 2}$

Step 11.2.3.1.3

Subtract $192$ from $1$ .

$x = \frac{1 \pm \sqrt{- 191}}{2 \cdot 2}$

Step 11.2.3.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 2}$

Step 11.2.3.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 2}$

Step 11.2.3.1.6

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

$x = \frac{1 \pm i \sqrt{191}}{2 \cdot 2}$

Step 11.2.3.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm i \sqrt{191}}{4}$

Step 11.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 11.2.4.1

Simplify the numerator.

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Step 11.2.4.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot 24}}{2 \cdot 2}$

Step 11.2.4.1.2

Multiply $- 4 \cdot 2 \cdot 24$ .

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Step 11.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot 24}}{2 \cdot 2}$

Step 11.2.4.1.2.2

Multiply $- 8$ by $24$ .

$x = \frac{1 \pm \sqrt{1 - 192}}{2 \cdot 2}$

Step 11.2.4.1.3

Subtract $192$ from $1$ .

$x = \frac{1 \pm \sqrt{- 191}}{2 \cdot 2}$

Step 11.2.4.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 2}$

Step 11.2.4.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 2}$

Step 11.2.4.1.6

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

$x = \frac{1 \pm i \sqrt{191}}{2 \cdot 2}$

Step 11.2.4.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm i \sqrt{191}}{4}$

Step 11.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{191}}{4}$

Step 11.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 11.2.5.1

Simplify the numerator.

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Step 11.2.5.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot 24}}{2 \cdot 2}$

Step 11.2.5.1.2

Multiply $- 4 \cdot 2 \cdot 24$ .

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Step 11.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot 24}}{2 \cdot 2}$

Step 11.2.5.1.2.2

Multiply $- 8$ by $24$ .

$x = \frac{1 \pm \sqrt{1 - 192}}{2 \cdot 2}$

Step 11.2.5.1.3

Subtract $192$ from $1$ .

$x = \frac{1 \pm \sqrt{- 191}}{2 \cdot 2}$

Step 11.2.5.1.4

Rewrite $- 191$ as $- 1 (191)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 191}}{2 \cdot 2}$

Step 11.2.5.1.5

Rewrite $\sqrt{- 1 (191)}$ as $\sqrt{- 1} \cdot \sqrt{191}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{191}}{2 \cdot 2}$

Step 11.2.5.1.6

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

$x = \frac{1 \pm i \sqrt{191}}{2 \cdot 2}$

Step 11.2.5.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm i \sqrt{191}}{4}$

Step 11.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{191}}{4}$

Step 11.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{191}}{4}, \frac{1 - i \sqrt{191}}{4}$

Step 12

The final solution is all the values that make $2 x (2 x^{2} - x + 24) = 0$ true.

$x = 0, \frac{1 + i \sqrt{191}}{4}, \frac{1 - i \sqrt{191}}{4}$

Step 13