Precalculus Examples

$3 X^{3} + 9 X^{2} + 8 X + 4 = X + 2$

Step 1

Move all terms containing $X$ to the left side of the equation.

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Step 1.1

Subtract $X$ from both sides of the equation.

$3 X^{3} + 9 X^{2} + 8 X + 4 - X = 2$

Step 1.2

Subtract $X$ from $8 X$ .

$3 X^{3} + 9 X^{2} + 7 X + 4 = 2$

Step 2

Subtract $2$ from both sides of the equation.

$3 X^{3} + 9 X^{2} + 7 X + 4 - 2 = 0$

Step 3

Subtract $2$ from $4$ .

$3 X^{3} + 9 X^{2} + 7 X + 2 = 0$

Step 4

Factor $3 X^{3} + 9 X^{2} + 7 X + 2$ using the rational roots test.

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Step 4.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 3$

Step 4.2

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

$\pm 1, \pm 0.\overline{3}, \pm 2, \pm 0.\overline{6}$

Step 4.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 4.3.1

Substitute $- 2$ into the polynomial.

$3 {(- 2)}^{3} + 9 {(- 2)}^{2} + 7 \cdot - 2 + 2$

Step 4.3.2

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

$3 \cdot - 8 + 9 {(- 2)}^{2} + 7 \cdot - 2 + 2$

Step 4.3.3

Multiply $3$ by $- 8$ .

$- 24 + 9 {(- 2)}^{2} + 7 \cdot - 2 + 2$

Step 4.3.4

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

$- 24 + 9 \cdot 4 + 7 \cdot - 2 + 2$

Step 4.3.5

Multiply $9$ by $4$ .

$- 24 + 36 + 7 \cdot - 2 + 2$

Step 4.3.6

Add $- 24$ and $36$ .

$12 + 7 \cdot - 2 + 2$

Step 4.3.7

Multiply $7$ by $- 2$ .

$12 - 14 + 2$

Step 4.3.8

Subtract $14$ from $12$ .

$- 2 + 2$

Step 4.3.9

Add $- 2$ and $2$ .

$0$

Step 4.4

Since $- 2$ is a known root, divide the polynomial by $X + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 X^{3} + 9 X^{2} + 7 X + 2}{X + 2}$

Step 4.5

Divide $3 X^{3} + 9 X^{2} + 7 X + 2$ by $X + 2$ .

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Step 4.5.1

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

$X$

$2$

$3 X^{3}$

$9 X^{2}$

$7 X$

$2$

Step 4.5.2

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

					$3 X^{2}$
	$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$

Step 4.5.3

Multiply the new quotient term by the divisor.

				$3 X^{2}$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			+	$3 X^{3}$	+	$6 X^{2}$

Step 4.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 X^{3} + 6 X^{2}$

				$3 X^{2}$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$

Step 4.5.5

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

				$3 X^{2}$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$

Step 4.5.6

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

				$3 X^{2}$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$

Step 4.5.7

Divide the highest order term in the dividend $3 X^{2}$ by the highest order term in divisor $X$ .

				$3 X^{2}$	+	$3 X$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$

Step 4.5.8

Multiply the new quotient term by the divisor.

				$3 X^{2}$	+	$3 X$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$
					+	$3 X^{2}$	+	$6 X$

Step 4.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 X^{2} + 6 X$

				$3 X^{2}$	+	$3 X$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$
					-	$3 X^{2}$	-	$6 X$

Step 4.5.10

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

				$3 X^{2}$	+	$3 X$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$
					-	$3 X^{2}$	-	$6 X$
							+	$X$

Step 4.5.11

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

				$3 X^{2}$	+	$3 X$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$
					-	$3 X^{2}$	-	$6 X$
							+	$X$	+	$2$

Step 4.5.12

Divide the highest order term in the dividend $X$ by the highest order term in divisor $X$ .

				$3 X^{2}$	+	$3 X$	+	$1$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$
					-	$3 X^{2}$	-	$6 X$
							+	$X$	+	$2$

Step 4.5.13

Multiply the new quotient term by the divisor.

				$3 X^{2}$	+	$3 X$	+	$1$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$
					-	$3 X^{2}$	-	$6 X$
							+	$X$	+	$2$
							+	$X$	+	$2$

Step 4.5.14

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

				$3 X^{2}$	+	$3 X$	+	$1$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$
					-	$3 X^{2}$	-	$6 X$
							+	$X$	+	$2$
							-	$X$	-	$2$

Step 4.5.15

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

				$3 X^{2}$	+	$3 X$	+	$1$
$X$	+	$2$		$3 X^{3}$	+	$9 X^{2}$	+	$7 X$	+	$2$
			-	$3 X^{3}$	-	$6 X^{2}$
					+	$3 X^{2}$	+	$7 X$
					-	$3 X^{2}$	-	$6 X$
							+	$X$	+	$2$
							-	$X$	-	$2$
										$0$

Step 4.5.16

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

$3 X^{2} + 3 X + 1$

Step 4.6

Write $3 X^{3} + 9 X^{2} + 7 X + 2$ as a set of factors.

$(X + 2) (3 X^{2} + 3 X + 1) = 0$

Step 5

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

$X + 2 = 0$

$3 X^{2} + 3 X + 1 = 0$

Step 6

Set $X + 2$ equal to $0$ and solve for $X$ .

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Step 6.1

Set $X + 2$ equal to $0$ .

$X + 2 = 0$

Step 6.2

Subtract $2$ from both sides of the equation.

$X = - 2$

Step 7

Set $3 X^{2} + 3 X + 1$ equal to $0$ and solve for $X$ .

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Step 7.1

Set $3 X^{2} + 3 X + 1$ equal to $0$ .

$3 X^{2} + 3 X + 1 = 0$

Step 7.2

Solve $3 X^{2} + 3 X + 1 = 0$ for $X$ .

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Step 7.2.1

Use the quadratic formula to find the solutions.

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

Step 7.2.2

Substitute the values $a = 3$ , $b = 3$ , and $c = 1$ into the quadratic formula and solve for $X$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (3 \cdot 1)}}{2 \cdot 3}$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Simplify the numerator.

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Step 7.2.3.1.1

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

$X = \frac{- 3 \pm \sqrt{9 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

Step 7.2.3.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 7.2.3.1.2.1

Multiply $- 4$ by $3$ .

$X = \frac{- 3 \pm \sqrt{9 - 12 \cdot 1}}{2 \cdot 3}$

Step 7.2.3.1.2.2

Multiply $- 12$ by $1$ .

$X = \frac{- 3 \pm \sqrt{9 - 12}}{2 \cdot 3}$

Step 7.2.3.1.3

Subtract $12$ from $9$ .

$X = \frac{- 3 \pm \sqrt{- 3}}{2 \cdot 3}$

Step 7.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$X = \frac{- 3 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 3}$

Step 7.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$X = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 3}$

Step 7.2.3.1.6

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

$X = \frac{- 3 \pm i \sqrt{3}}{2 \cdot 3}$

Step 7.2.3.2

Multiply $2$ by $3$ .

$X = \frac{- 3 \pm i \sqrt{3}}{6}$

Step 7.2.4

The final answer is the combination of both solutions.

$X = - \frac{3 - i \sqrt{3}}{6}, - \frac{3 + i \sqrt{3}}{6}$

Step 8

The final solution is all the values that make $(X + 2) (3 X^{2} + 3 X + 1) = 0$ true.

$X = - 2, - \frac{3 - i \sqrt{3}}{6}, - \frac{3 + i \sqrt{3}}{6}$