Precalculus Examples

$f (x) = x^{3} - x^{2} + 4 x - 4$

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.

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

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 4$

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.

${(1)}^{3} - {(1)}^{2} + 4 (1) - 4$

Step 4

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

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

$1 - {(1)}^{2} + 4 (1) - 4$

Step 4.1.2

One to any power is one.

$1 - 1 \cdot 1 + 4 (1) - 4$

Step 4.1.3

Multiply $- 1$ by $1$ .

$1 - 1 + 4 (1) - 4$

Step 4.1.4

Multiply $4$ by $1$ .

$1 - 1 + 4 - 4$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $1$ from $1$ .

$0 + 4 - 4$

Step 4.2.2

Add $0$ and $4$ .

$4 - 4$

Step 4.2.3

Subtract $4$ from $4$ .

$0$

Step 5

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

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

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.

$1$	$1$	$- 1$	$4$	$- 4$

Step 6.2

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

$1$	$1$	$- 1$	$4$	$- 4$

	$1$

Step 6.3

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

$1$	$1$	$- 1$	$4$	$- 4$
		$1$
	$1$

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.

$1$	$1$	$- 1$	$4$	$- 4$
		$1$
	$1$	$0$

Step 6.5

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

$1$	$1$	$- 1$	$4$	$- 4$
		$1$	$0$
	$1$	$0$

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.

$1$	$1$	$- 1$	$4$	$- 4$
		$1$	$0$
	$1$	$0$	$4$

Step 6.7

Multiply the newest entry in the result $(4)$ by the divisor $(1)$ and place the result of $(4)$ under the next term in the dividend $(- 4)$ .

$1$	$1$	$- 1$	$4$	$- 4$
		$1$	$0$	$4$
	$1$	$0$	$4$

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.

$1$	$1$	$- 1$	$4$	$- 4$
		$1$	$0$	$4$
	$1$	$0$	$4$	$0$

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.

$1 x^{2} + 0 x + 4$

Step 6.10

Simplify the quotient polynomial.

$x^{2} + 4$

Step 7

Solve the equation to find any remaining roots.

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Step 7.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

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

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

Step 7.3.4

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

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

Step 7.3.5

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

$x = \pm i \cdot 2$

Step 7.3.6

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

$x = \pm 2 i$

Step 7.4

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

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Step 7.4.1

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

$x = 2 i$

Step 7.4.2

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

$x = - 2 i$

Step 7.4.3

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

$x = 2 i, - 2 i$

Step 8

The polynomial can be written as a set of linear factors.

$(x - 1) (x - 2 i) (x + 2 i)$

Step 9

These are the roots (zeros) of the polynomial $x^{3} - x^{2} + 4 x - 4$ .

$x = 1, 2 i, - 2 i$

Step 10