Trigonometry Examples

$g (x) = 3 x^{3} + 2 x^{2} - 7 x - 6$

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.

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

$q = \pm 1, \pm 3$

Step 2

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

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

Step 3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 3.1

Substitute $- 1$ into the polynomial.

$3 {(- 1)}^{3} + 2 {(- 1)}^{2} - 7 \cdot - 1 - 6$

Step 3.2

Raise $- 1$ to the power of $3$ .

$3 \cdot - 1 + 2 {(- 1)}^{2} - 7 \cdot - 1 - 6$

Step 3.3

Multiply $3$ by $- 1$ .

$- 3 + 2 {(- 1)}^{2} - 7 \cdot - 1 - 6$

Step 3.4

Raise $- 1$ to the power of $2$ .

$- 3 + 2 \cdot 1 - 7 \cdot - 1 - 6$

Step 3.5

Multiply $2$ by $1$ .

$- 3 + 2 - 7 \cdot - 1 - 6$

Step 3.6

Add $- 3$ and $2$ .

$- 1 - 7 \cdot - 1 - 6$

Step 3.7

Multiply $- 7$ by $- 1$ .

$- 1 + 7 - 6$

Step 3.8

Add $- 1$ and $7$ .

$6 - 6$

Step 3.9

Subtract $6$ from $6$ .

$0$

Step 4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 5

Divide $3 x^{3} + 2 x^{2} - 7 x - 6$ by $x + 1$ .

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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 $0$ .

$x$

$1$

$3 x^{3}$

$2 x^{2}$

$7 x$

$6$

Step 5.2

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

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

Step 5.3

Multiply the new quotient term by the divisor.

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Multiply the new quotient term by the divisor.

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

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

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

Step 5.12

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

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

Step 5.13

Multiply the new quotient term by the divisor.

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

Step 5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x - 6$

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

Step 5.15

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

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

Step 5.16

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

$3 x^{2} - x - 6$

Step 6

Write $3 x^{3} + 2 x^{2} - 7 x - 6$ as a set of factors.

$g (x) = (x + 1) (3 x^{2} - x - 6)$