Precalculus Examples

$3 x^{3} + 4 x^{2} - 7 x + 2$

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 3$

Step 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 3

Substitute $0.\overline{6}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{6}$ is a root of the polynomial.

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Step 3.1

Substitute $0.\overline{6}$ into the polynomial.

$3 \cdot {0.\overline{6}}^{3} + 4 \cdot {0.\overline{6}}^{2} - 7 \cdot 0.\overline{6} + 2$

Step 3.2

Raise $0.\overline{6}$ to the power of $3$ .

$3 \cdot 0.\overline{296} + 4 \cdot {0.\overline{6}}^{2} - 7 \cdot 0.\overline{6} + 2$

Step 3.3

Multiply $3$ by $0.\overline{296}$ .

$0.\overline{8} + 4 \cdot {0.\overline{6}}^{2} - 7 \cdot 0.\overline{6} + 2$

Step 3.4

Raise $0.\overline{6}$ to the power of $2$ .

$0.\overline{8} + 4 \cdot 0.\overline{4} - 7 \cdot 0.\overline{6} + 2$

Step 3.5

Multiply $4$ by $0.\overline{4}$ .

$0.\overline{8} + 1.\overline{7} - 7 \cdot 0.\overline{6} + 2$

Step 3.6

Add $0.\overline{8}$ and $1.\overline{7}$ .

$2.\overline{6} - 7 \cdot 0.\overline{6} + 2$

Step 3.7

Multiply $- 7$ by $0.\overline{6}$ .

$2.\overline{6} - 4.\overline{6} + 2$

Step 3.8

Subtract $4.\overline{6}$ from $2.\overline{6}$ .

$- 2 + 2$

Step 3.9

Add $- 2$ and $2$ .

$0$

Step 4

Since $0.\overline{6}$ is a known root, divide the polynomial by $3 x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{3} + 4 x^{2} - 7 x + 2}{3 x - 2}$

Step 5

Divide $3 x^{3} + 4 x^{2} - 7 x + 2$ by $3 x - 2$ .

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Step 5.1

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

$3 x$

$2$

$3 x^{3}$

$4 x^{2}$

$7 x$

$2$

Step 5.2

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

					$x^{2}$
	$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$

Step 5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			+	$3 x^{3}$	-	$2 x^{2}$

Step 5.4

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

				$x^{2}$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$

Step 5.5

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

				$x^{2}$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$

Step 5.6

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

				$x^{2}$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$

Step 5.7

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $3 x$ .

				$x^{2}$	+	$2 x$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$

Step 5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					+	$6 x^{2}$	-	$4 x$

Step 5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} - 4 x$

				$x^{2}$	+	$2 x$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$4 x$

Step 5.10

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

				$x^{2}$	+	$2 x$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$4 x$
							-	$3 x$

Step 5.11

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

				$x^{2}$	+	$2 x$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$4 x$
							-	$3 x$	+	$2$

Step 5.12

Divide the highest order term in the dividend $- 3 x$ by the highest order term in divisor $3 x$ .

				$x^{2}$	+	$2 x$	-	$1$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$4 x$
							-	$3 x$	+	$2$

Step 5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$1$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$4 x$
							-	$3 x$	+	$2$
							-	$3 x$	+	$2$

Step 5.14

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

				$x^{2}$	+	$2 x$	-	$1$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$4 x$
							-	$3 x$	+	$2$
							+	$3 x$	-	$2$

Step 5.15

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

				$x^{2}$	+	$2 x$	-	$1$
$3 x$	-	$2$		$3 x^{3}$	+	$4 x^{2}$	-	$7 x$	+	$2$
			-	$3 x^{3}$	+	$2 x^{2}$
					+	$6 x^{2}$	-	$7 x$
					-	$6 x^{2}$	+	$4 x$
							-	$3 x$	+	$2$
							+	$3 x$	-	$2$
										$0$

Step 5.16

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

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

Step 6

Write $3 x^{3} + 4 x^{2} - 7 x + 2$ as a set of factors.

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