Precalculus Examples

$2 x^{3} - 3 x^{2} - 4 x + 6 = 0$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 6$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm 6$

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.

$2 {(\frac{3}{2})}^{3} - 3 {(\frac{3}{2})}^{2} - 4 (\frac{3}{2}) + 6$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = \frac{3}{2}$ 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 $\frac{3}{2}$ .

$2 \frac{3^{3}}{2^{3}} - 3 {(\frac{3}{2})}^{2} - 4 (\frac{3}{2}) + 6$

Step 4.1.2

Raise $3$ to the power of $3$ .

$2 \frac{27}{2^{3}} - 3 {(\frac{3}{2})}^{2} - 4 (\frac{3}{2}) + 6$

Step 4.1.3

Raise $2$ to the power of $3$ .

$2 (\frac{27}{8}) - 3 {(\frac{3}{2})}^{2} - 4 (\frac{3}{2}) + 6$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Factor $2$ out of $8$ .

$2 \frac{27}{2 (4)} - 3 {(\frac{3}{2})}^{2} - 4 (\frac{3}{2}) + 6$

Step 4.1.4.2

Cancel the common factor.

$2 \frac{27}{2 \cdot 4} - 3 {(\frac{3}{2})}^{2} - 4 (\frac{3}{2}) + 6$

Step 4.1.4.3

Rewrite the expression.

$\frac{27}{4} - 3 {(\frac{3}{2})}^{2} - 4 (\frac{3}{2}) + 6$

Step 4.1.5

Apply the product rule to $\frac{3}{2}$ .

$\frac{27}{4} - 3 \frac{3^{2}}{2^{2}} - 4 (\frac{3}{2}) + 6$

Step 4.1.6

Raise $3$ to the power of $2$ .

$\frac{27}{4} - 3 \frac{9}{2^{2}} - 4 (\frac{3}{2}) + 6$

Step 4.1.7

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

$\frac{27}{4} - 3 (\frac{9}{4}) - 4 (\frac{3}{2}) + 6$

Step 4.1.8

Multiply $- 3 (\frac{9}{4})$ .

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Step 4.1.8.1

Combine $- 3$ and $\frac{9}{4}$ .

$\frac{27}{4} + \frac{- 3 \cdot 9}{4} - 4 (\frac{3}{2}) + 6$

Step 4.1.8.2

Multiply $- 3$ by $9$ .

$\frac{27}{4} + \frac{- 27}{4} - 4 (\frac{3}{2}) + 6$

Step 4.1.9

Move the negative in front of the fraction.

$\frac{27}{4} - \frac{27}{4} - 4 (\frac{3}{2}) + 6$

Step 4.1.10

Cancel the common factor of $2$ .

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Step 4.1.10.1

Factor $2$ out of $- 4$ .

$\frac{27}{4} - \frac{27}{4} + 2 (- 2) \frac{3}{2} + 6$

Step 4.1.10.2

Cancel the common factor.

$\frac{27}{4} - \frac{27}{4} + 2 \cdot - 2 \frac{3}{2} + 6$

Step 4.1.10.3

Rewrite the expression.

$\frac{27}{4} - \frac{27}{4} - 2 \cdot 3 + 6$

Step 4.1.11

Multiply $- 2$ by $3$ .

$\frac{27}{4} - \frac{27}{4} - 6 + 6$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$- 6 + 6 + \frac{27 - 27}{4}$

Step 4.2.2

Subtract $27$ from $27$ .

$- 6 + 6 + \frac{0}{4}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $- 6$ as a fraction with denominator $1$ .

$\frac{- 6}{1} + 6 + \frac{0}{4}$

Step 4.3.2

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$\frac{- 6}{1} \cdot \frac{4}{4} + 6 + \frac{0}{4}$

Step 4.3.3

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$\frac{- 6 \cdot 4}{4} + 6 + \frac{0}{4}$

Step 4.3.4

Write $6$ as a fraction with denominator $1$ .

$\frac{- 6 \cdot 4}{4} + \frac{6}{1} + \frac{0}{4}$

Step 4.3.5

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$\frac{- 6 \cdot 4}{4} + \frac{6}{1} \cdot \frac{4}{4} + \frac{0}{4}$

Step 4.3.6

Multiply $\frac{6}{1}$ by $\frac{4}{4}$ .

$\frac{- 6 \cdot 4}{4} + \frac{6 \cdot 4}{4} + \frac{0}{4}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{- 6 \cdot 4 + 6 \cdot 4}{4}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $- 6$ by $4$ .

$\frac{- 24 + 6 \cdot 4}{4}$

Step 4.5.2

Multiply $6$ by $4$ .

$\frac{- 24 + 24}{4}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $- 24$ and $24$ .

$\frac{0}{4}$

Step 4.6.2

Divide $0$ by $4$ .

$0$

Step 5

Since $\frac{3}{2}$ is a known root, divide the polynomial by $x - \frac{3}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - 3 x^{2} - 4 x + 6}{x - \frac{3}{2}}$

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.

$\frac{3}{2}$	$2$	$- 3$	$- 4$	$6$

Step 6.2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$\frac{3}{2}$	$2$	$- 3$	$- 4$	$6$

	$2$

Step 6.3

Multiply the newest entry in the result $(2)$ by the divisor $(\frac{3}{2})$ and place the result of $(3)$ under the next term in the dividend $(- 3)$ .

$\frac{3}{2}$	$2$	$- 3$	$- 4$	$6$
		$3$
	$2$

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.

$\frac{3}{2}$	$2$	$- 3$	$- 4$	$6$
		$3$
	$2$	$0$

Step 6.5

Multiply the newest entry in the result $(0)$ by the divisor $(\frac{3}{2})$ and place the result of $(0)$ under the next term in the dividend $(- 4)$ .

$\frac{3}{2}$	$2$	$- 3$	$- 4$	$6$
		$3$	$0$
	$2$	$0$

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.

$\frac{3}{2}$	$2$	$- 3$	$- 4$	$6$
		$3$	$0$
	$2$	$0$	$- 4$

Step 6.7

Multiply the newest entry in the result $(- 4)$ by the divisor $(\frac{3}{2})$ and place the result of $(- 6)$ under the next term in the dividend $(6)$ .

$\frac{3}{2}$	$2$	$- 3$	$- 4$	$6$
		$3$	$0$	$- 6$
	$2$	$0$	$- 4$

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.

$\frac{3}{2}$	$2$	$- 3$	$- 4$	$6$
		$3$	$0$	$- 6$
	$2$	$0$	$- 4$	$0$

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.

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

Step 6.10

Simplify the quotient polynomial.

$2 x^{2} - 4$

Step 7

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

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Step 7.1

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

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

Step 7.2

Factor $2$ out of $- 4$ .

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

Step 7.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 2$ .

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

Step 8

Factor the left side of the equation.

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Step 8.1

Factor out the greatest common factor from each group.

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Step 8.1.1

Group the first two terms and the last two terms.

$(2 x^{3} - 3 x^{2}) - 4 x + 6 = 0$

Step 8.1.2

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

$x^{2} (2 x - 3) - 2 (2 x - 3) = 0$

Step 8.2

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$(2 x - 3) (x^{2} - 2) = 0$

Step 9

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

$2 x - 3 = 0$

$x^{2} - 2 = 0$

Step 10

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 10.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 10.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 10.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 10.2.2.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $2$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 11

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

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Step 11.1

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

$x^{2} - 2 = 0$

Step 11.2

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

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Step 11.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 11.2.2

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

$x = \pm \sqrt{2}$

Step 11.2.3

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

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Step 11.2.3.1

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

$x = \sqrt{2}$

Step 11.2.3.2

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

$x = - \sqrt{2}$

Step 11.2.3.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 12

The final solution is all the values that make $(2 x - 3) (x^{2} - 2) = 0$ true.

$x = \frac{3}{2}, \sqrt{2}, - \sqrt{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{2}, \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = 1.5, 1.41421356 \dots, - 1.41421356 \dots$

Mixed Number Form:

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

Step 14