Precalculus Examples

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

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.

$p = \pm 1$

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

Step 2

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

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

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.

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

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.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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

Step 4.1.2

One to any power is one.

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

Step 4.1.3

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

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

Step 4.1.4

Cancel the common factor of $4$ .

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Step 4.1.4.1

Factor $4$ out of $- 4$ .

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

Step 4.1.4.2

Cancel the common factor.

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

Step 4.1.4.3

Rewrite the expression.

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

Step 4.1.5

Cancel the common factor of $2$ .

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Step 4.1.5.1

Factor $2$ out of $4$ .

$- 1 + 2 (2) \frac{1}{2} - 1$

Step 4.1.5.2

Cancel the common factor.

$- 1 + 2 \cdot 2 \frac{1}{2} - 1$

Step 4.1.5.3

Rewrite the expression.

$- 1 + 2 - 1$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Add $- 1$ and $2$ .

$1 - 1$

Step 4.2.2

Subtract $1$ from $1$ .

$0$

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.

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

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.

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

Step 6.2

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

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

	$- 4$

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 $(4)$ .

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

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.

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

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 $(- 1)$ .

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

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.

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

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.

$(- 4) x + 2$

Step 6.8

Simplify the quotient polynomial.

$- 4 x + 2$

Step 7

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

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Step 7.1

Factor $2$ out of $- 4 x$ .

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

Step 7.2

Factor $2$ out of $2$ .

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

Step 7.3

Factor $2$ out of $2 (- 2 x) + 2 (1)$ .

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

Step 8

Factor the left side of the equation.

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Step 8.1

Factor $- 1$ out of $- 4 x^{2} + 4 x - 1$ .

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Step 8.1.1

Factor $- 1$ out of $- 4 x^{2}$ .

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

Step 8.1.2

Factor $- 1$ out of $4 x$ .

$- (4 x^{2}) - (- 4 x) - 1 = 0$

Step 8.1.3

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

$- (4 x^{2}) - (- 4 x) - 1 \cdot 1 = 0$

Step 8.1.4

Factor $- 1$ out of $- (4 x^{2}) - (- 4 x)$ .

$- (4 x^{2} - 4 x) - 1 \cdot 1 = 0$

Step 8.1.5

Factor $- 1$ out of $- (4 x^{2} - 4 x) - 1 (1)$ .

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

Step 8.2

Factor using the perfect square rule.

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Step 8.2.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 8.2.2

Rewrite $1$ as $1^{2}$ .

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

Step 8.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot (2 x) \cdot 1$

Step 8.2.4

Rewrite the polynomial.

$- ({(2 x)}^{2} - 2 \cdot (2 x) \cdot 1 + 1^{2}) = 0$

Step 8.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 x$ and $b = 1$ .

$- {(2 x - 1)}^{2} = 0$

Step 9

Divide each term in $- {(2 x - 1)}^{2} = 0$ by $- 1$ and simplify.

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Step 9.1

Divide each term in $- {(2 x - 1)}^{2} = 0$ by $- 1$ .

$\frac{- {(2 x - 1)}^{2}}{- 1} = \frac{0}{- 1}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Dividing two negative values results in a positive value.

$\frac{{(2 x - 1)}^{2}}{1} = \frac{0}{- 1}$

Step 9.2.2

Divide ${(2 x - 1)}^{2}$ by $1$ .

${(2 x - 1)}^{2} = \frac{0}{- 1}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Divide $0$ by $- 1$ .

${(2 x - 1)}^{2} = 0$

Step 10

Set the $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 11

Solve for $x$ .

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Step 11.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 11.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 11.2.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 11.2.2

Simplify the left side.

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Step 11.2.2.1

Cancel the common factor of $2$ .

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Step 11.2.2.1.1

Cancel the common factor.

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

Step 11.2.2.1.2

Divide $x$ by $1$ .

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

Step 12