Precalculus Examples

$7 x^{3} + 12 x^{2} - 11 x + 2 = 0$

Step 1

Factor $7 x^{3} + 12 x^{2} - 11 x + 2$ using the rational roots test.

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Step 1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 7$

Step 1.2

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

$\pm 1, \pm 0.\overline{142857}, \pm 2, \pm 0.\overline{285714}$

Step 1.3

Substitute $0.\overline{285714}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{285714}$ is a root of the polynomial.

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Step 1.3.1

Substitute $0.\overline{285714}$ into the polynomial.

$7 \cdot {0.\overline{285714}}^{3} + 12 \cdot {0.\overline{285714}}^{2} - 11 \cdot 0.\overline{285714} + 2$

Step 1.3.2

Raise $0.\overline{285714}$ to the power of $3$ .

$7 \cdot 0.02332361 + 12 \cdot {0.\overline{285714}}^{2} - 11 \cdot 0.\overline{285714} + 2$

Step 1.3.3

Multiply $7$ by $0.02332361$ .

$0.1632653 + 12 \cdot {0.\overline{285714}}^{2} - 11 \cdot 0.\overline{285714} + 2$

Step 1.3.4

Raise $0.\overline{285714}$ to the power of $2$ .

$0.1632653 + 12 \cdot 0.08163265 - 11 \cdot 0.\overline{285714} + 2$

Step 1.3.5

Multiply $12$ by $0.08163265$ .

$0.1632653 + 0.97959183 - 11 \cdot 0.\overline{285714} + 2$

Step 1.3.6

Add $0.1632653$ and $0.97959183$ .

$1.14285714 - 11 \cdot 0.\overline{285714} + 2$

Step 1.3.7

Multiply $- 11$ by $0.\overline{285714}$ .

$1.14285714 - 3.\overline{142857} + 2$

Step 1.3.8

Subtract $3.\overline{142857}$ from $1.14285714$ .

$- 2 + 2$

Step 1.3.9

Add $- 2$ and $2$ .

$0$

Step 1.4

Since $0.\overline{285714}$ is a known root, divide the polynomial by $7 x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{7 x^{3} + 12 x^{2} - 11 x + 2}{7 x - 2}$

Step 1.5

Divide $7 x^{3} + 12 x^{2} - 11 x + 2$ by $7 x - 2$ .

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Step 1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$7 x$

$2$

$7 x^{3}$

$12 x^{2}$

$11 x$

$2$

Step 1.5.2

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $7 x$ .

					$x^{2}$
	$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			+	$7 x^{3}$	-	$2 x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} - 2 x^{2}$

				$x^{2}$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$

Step 1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$

Step 1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$

Step 1.5.7

Divide the highest order term in the dividend $14 x^{2}$ by the highest order term in divisor $7 x$ .

				$x^{2}$	+	$2 x$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$
					+	$14 x^{2}$	-	$4 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $14 x^{2} - 4 x$

				$x^{2}$	+	$2 x$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$
					-	$14 x^{2}$	+	$4 x$

Step 1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$2 x$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$
					-	$14 x^{2}$	+	$4 x$
							-	$7 x$

Step 1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$2 x$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$
					-	$14 x^{2}$	+	$4 x$
							-	$7 x$	+	$2$

Step 1.5.12

Divide the highest order term in the dividend $- 7 x$ by the highest order term in divisor $7 x$ .

				$x^{2}$	+	$2 x$	-	$1$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$
					-	$14 x^{2}$	+	$4 x$
							-	$7 x$	+	$2$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$1$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$
					-	$14 x^{2}$	+	$4 x$
							-	$7 x$	+	$2$
							-	$7 x$	+	$2$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x + 2$

				$x^{2}$	+	$2 x$	-	$1$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$
					-	$14 x^{2}$	+	$4 x$
							-	$7 x$	+	$2$
							+	$7 x$	-	$2$

Step 1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$2 x$	-	$1$
$7 x$	-	$2$		$7 x^{3}$	+	$12 x^{2}$	-	$11 x$	+	$2$
			-	$7 x^{3}$	+	$2 x^{2}$
					+	$14 x^{2}$	-	$11 x$
					-	$14 x^{2}$	+	$4 x$
							-	$7 x$	+	$2$
							+	$7 x$	-	$2$
										$0$

Step 1.5.16

Since the remander is $0$ , the final answer is the quotient.

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

Step 1.6

Write $7 x^{3} + 12 x^{2} - 11 x + 2$ as a set of factors.

$(7 x - 2) (x^{2} + 2 x - 1) = 0$

Step 2

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

$7 x - 2 = 0$

$x^{2} + 2 x - 1 = 0$

Step 3

Set $7 x - 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $7 x - 2$ equal to $0$ .

$7 x - 2 = 0$

Step 3.2

Solve $7 x - 2 = 0$ for $x$ .

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Step 3.2.1

Add $2$ to both sides of the equation.

$7 x = 2$

Step 3.2.2

Divide each term in $7 x = 2$ by $7$ and simplify.

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Step 3.2.2.1

Divide each term in $7 x = 2$ by $7$ .

$\frac{7 x}{7} = \frac{2}{7}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $7$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{2}{7}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{7}$

Step 4

Set $x^{2} + 2 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} + 2 x - 1$ equal to $0$ .

$x^{2} + 2 x - 1 = 0$

Step 4.2

Solve $x^{2} + 2 x - 1 = 0$ for $x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

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

Step 4.2.2

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

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

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

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

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

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.2.3.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 4.2.3.1.3

Add $4$ and $4$ .

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

Step 4.2.3.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.2.3.1.4.1

Factor $4$ out of $8$ .

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

Step 4.2.3.1.4.2

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

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

Step 4.2.3.1.5

Pull terms out from under the radical.

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

Step 4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{2}}{2}$

Step 4.2.3.3

Simplify $\frac{- 2 \pm 2 \sqrt{2}}{2}$ .

$x = - 1 \pm \sqrt{2}$

Step 4.2.4

The final answer is the combination of both solutions.

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

Step 5

The final solution is all the values that make $(7 x - 2) (x^{2} + 2 x - 1) = 0$ true.

$x = \frac{2}{7}, - 1 + \sqrt{2}, - 1 - \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{7}, - 1 + \sqrt{2}, - 1 - \sqrt{2}$

Decimal Form:

$x = 0.\overline{285714}, 0.41421356 \dots, - 2.41421356 \dots$