Trigonometry Examples

$f (x) = x^{4} - 2 x^{3} - 7 x^{2} - 8 + 12$

Step 1

Add $- 8$ and $12$ .

$f (x) = x^{4} - 2 x^{3} - 7 x^{2} + 4$

Step 2

Factor $x^{4} - 2 x^{3} - 7 x^{2} + 4$ using the rational roots test.

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Step 2.1

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.

$p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

Step 2.2

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

$\pm 1, \pm 4, \pm 2$

Step 2.3

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 2.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - 2 {(- 1)}^{3} - 7 {(- 1)}^{2} + 4$

Step 2.3.2

Raise $- 1$ to the power of $4$ .

$1 - 2 {(- 1)}^{3} - 7 {(- 1)}^{2} + 4$

Step 2.3.3

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

$1 - 2 \cdot - 1 - 7 {(- 1)}^{2} + 4$

Step 2.3.4

Multiply $- 2$ by $- 1$ .

$1 + 2 - 7 {(- 1)}^{2} + 4$

Step 2.3.5

Add $1$ and $2$ .

$3 - 7 {(- 1)}^{2} + 4$

Step 2.3.6

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

$3 - 7 \cdot 1 + 4$

Step 2.3.7

Multiply $- 7$ by $1$ .

$3 - 7 + 4$

Step 2.3.8

Subtract $7$ from $3$ .

$- 4 + 4$

Step 2.3.9

Add $- 4$ and $4$ .

$0$

Step 2.4

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.

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

Step 2.5

Divide $x^{4} - 2 x^{3} - 7 x^{2} + 4$ by $x + 1$ .

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Step 2.5.1

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

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

Step 2.5.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

Step 2.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

Step 2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + x^{3}$

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

Step 2.5.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

Step 2.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

Step 2.5.7

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

Step 2.5.8

Multiply the new quotient term by the divisor.

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

Step 2.5.9

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

Step 2.5.10

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

Step 2.5.11

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

Step 2.5.12

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

Step 2.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

Step 2.5.14

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

$x^{3}$

$3 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

Step 2.5.15

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

$x^{3}$

$3 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

Step 2.5.16

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

$x^{3}$

$3 x^{2}$

$4 x$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

Step 2.5.17

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$4 x$

$4$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

Step 2.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$4 x$

$4$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

$4 x$

$4$

Step 2.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 4$

$x^{3}$

$3 x^{2}$

$4 x$

$4$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

$4 x$

$4$

Step 2.5.20

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

$x^{3}$

$3 x^{2}$

$4 x$

$4$

$x$

$1$

$x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$4$

$x^{4}$

$x^{3}$

$3 x^{3}$

$7 x^{2}$

$3 x^{3}$

$3 x^{2}$

$4 x^{2}$

$0 x$

$4 x^{2}$

$4 x$

$4$

$4 x$

$4$

$0$

Step 2.5.21

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

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

Step 2.6

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

$f (x) = (x + 1) (x^{3} - 3 x^{2} - 4 x + 4)$