Algebra Examples

$f (x) = \frac{x^{3} - 21 x + 20}{x^{3} - 8 x^{2} + 11 x + 20}$

Step 1

Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

Step 2

Find $f (- x)$ .

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Step 2.1

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = \frac{{(- x)}^{3} - 21 (- x) + 20}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = \frac{{(- 1)}^{3} x^{3} - 21 (- x) + 20}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2.2

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

$f (- x) = \frac{- x^{3} - 21 (- x) + 20}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2.3

Multiply $- 1$ by $- 21$ .

$f (- x) = \frac{- x^{3} + 21 x + 20}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2.4

Rewrite $- x^{3} + 21 x + 20$ in a factored form.

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Step 2.2.4.1

Factor $- x^{3} + 21 x + 20$ using the rational roots test.

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Step 2.2.4.1.1

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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

$q = \pm 1$

Step 2.2.4.1.2

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

$\pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

Step 2.2.4.1.3

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 2.2.4.1.3.1

Substitute $- 1$ into the polynomial.

$- {(- 1)}^{3} + 21 \cdot - 1 + 20$

Step 2.2.4.1.3.2

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

$- - 1 + 21 \cdot - 1 + 20$

Step 2.2.4.1.3.3

Multiply $- 1$ by $- 1$ .

$1 + 21 \cdot - 1 + 20$

Step 2.2.4.1.3.4

Multiply $21$ by $- 1$ .

$1 - 21 + 20$

Step 2.2.4.1.3.5

Subtract $21$ from $1$ .

$- 20 + 20$

Step 2.2.4.1.3.6

Add $- 20$ and $20$ .

$0$

Step 2.2.4.1.4

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.

$\frac{- x^{3} + 21 x + 20}{x + 1}$

Step 2.2.4.1.5

Divide $- x^{3} + 21 x + 20$ by $x + 1$ .

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Step 2.2.4.1.5.1

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

$x$

$1$

$x^{3}$

$0 x^{2}$

$21 x$

$20$

Step 2.2.4.1.5.2

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

				-	$x^{2}$
	$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$

Step 2.2.4.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			-	$x^{3}$	-	$x^{2}$

Step 2.2.4.1.5.4

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

			-	$x^{2}$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$

Step 2.2.4.1.5.5

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

			-	$x^{2}$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$

Step 2.2.4.1.5.6

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

			-	$x^{2}$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$

Step 2.2.4.1.5.7

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

			-	$x^{2}$	+	$x$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$

Step 2.2.4.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$x$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$
					+	$x^{2}$	+	$x$

Step 2.2.4.1.5.9

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

			-	$x^{2}$	+	$x$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$
					-	$x^{2}$	-	$x$

Step 2.2.4.1.5.10

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

			-	$x^{2}$	+	$x$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$
					-	$x^{2}$	-	$x$
							+	$20 x$

Step 2.2.4.1.5.11

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

			-	$x^{2}$	+	$x$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$
					-	$x^{2}$	-	$x$
							+	$20 x$	+	$20$

Step 2.2.4.1.5.12

Divide the highest order term in the dividend $20 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$x$	+	$20$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$
					-	$x^{2}$	-	$x$
							+	$20 x$	+	$20$

Step 2.2.4.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$x$	+	$20$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$
					-	$x^{2}$	-	$x$
							+	$20 x$	+	$20$
							+	$20 x$	+	$20$

Step 2.2.4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $20 x + 20$

			-	$x^{2}$	+	$x$	+	$20$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$
					-	$x^{2}$	-	$x$
							+	$20 x$	+	$20$
							-	$20 x$	-	$20$

Step 2.2.4.1.5.15

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

			-	$x^{2}$	+	$x$	+	$20$
$x$	+	$1$	-	$x^{3}$	+	$0 x^{2}$	+	$21 x$	+	$20$
			+	$x^{3}$	+	$x^{2}$
					+	$x^{2}$	+	$21 x$
					-	$x^{2}$	-	$x$
							+	$20 x$	+	$20$
							-	$20 x$	-	$20$
										$0$

Step 2.2.4.1.5.16

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

$- x^{2} + x + 20$

Step 2.2.4.1.6

Write $- x^{3} + 21 x + 20$ as a set of factors.

$f (- x) = \frac{(x + 1) (- x^{2} + x + 20)}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2.4.2

Factor by grouping.

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Step 2.2.4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 20 = - 20$ and whose sum is $b = 1$ .

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Step 2.2.4.2.1.1

Multiply by $1$ .

$f (- x) = \frac{(x + 1) (- x^{2} + 1 x + 20)}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2.4.2.1.2

Rewrite $1$ as $- 4$ plus $5$

$f (- x) = \frac{(x + 1) (- x^{2} + (- 4 + 5) x + 20)}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2.4.2.1.3

Apply the distributive property.

$f (- x) = \frac{(x + 1) (- x^{2} - 4 x + 5 x + 20)}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2.4.2.2

Factor out the greatest common factor from each group.

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Step 2.2.4.2.2.1

Group the first two terms and the last two terms.

$f (- x) = \frac{(x + 1) ((- x^{2} - 4 x) + 5 x + 20)}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2.4.2.2.2

Factor out the greatest common factor (GCF) from each group.

$f (- x) = \frac{(x + 1) (x (- x - 4) - 5 (- x - 4))}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.2.4.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 4$ .

$f (- x) = \frac{(x + 1) ((- x - 4) (x - 5))}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{{(- x)}^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Apply the product rule to $- x$ .

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{{(- 1)}^{3} x^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.3.2

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

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{- x^{3} - 8 {(- x)}^{2} + 11 (- x) + 20}$

Step 2.3.3

Apply the product rule to $- x$ .

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{- x^{3} - 8 ({(- 1)}^{2} x^{2}) + 11 (- x) + 20}$

Step 2.3.4

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

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{- x^{3} - 8 (1 x^{2}) + 11 (- x) + 20}$

Step 2.3.5

Multiply $x^{2}$ by $1$ .

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{- x^{3} - 8 x^{2} + 11 (- x) + 20}$

Step 2.3.6

Multiply $- 1$ by $11$ .

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{- x^{3} - 8 x^{2} - 11 x + 20}$

Step 2.3.7

Rewrite $- x^{3} - 8 x^{2} - 11 x + 20$ in a factored form.

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Step 2.3.7.1

Factor $- x^{3} - 8 x^{2} - 11 x + 20$ using the rational roots test.

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Step 2.3.7.1.1

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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

$q = \pm 1$

Step 2.3.7.1.2

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

$\pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

Step 2.3.7.1.3

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 2.3.7.1.3.1

Substitute $1$ into the polynomial.

$- 1^{3} - 8 \cdot 1^{2} - 11 \cdot 1 + 20$

Step 2.3.7.1.3.2

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

$- 1 \cdot 1 - 8 \cdot 1^{2} - 11 \cdot 1 + 20$

Step 2.3.7.1.3.3

Multiply $- 1$ by $1$ .

$- 1 - 8 \cdot 1^{2} - 11 \cdot 1 + 20$

Step 2.3.7.1.3.4

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

$- 1 - 8 \cdot 1 - 11 \cdot 1 + 20$

Step 2.3.7.1.3.5

Multiply $- 8$ by $1$ .

$- 1 - 8 - 11 \cdot 1 + 20$

Step 2.3.7.1.3.6

Subtract $8$ from $- 1$ .

$- 9 - 11 \cdot 1 + 20$

Step 2.3.7.1.3.7

Multiply $- 11$ by $1$ .

$- 9 - 11 + 20$

Step 2.3.7.1.3.8

Subtract $11$ from $- 9$ .

$- 20 + 20$

Step 2.3.7.1.3.9

Add $- 20$ and $20$ .

$0$

Step 2.3.7.1.4

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.

$\frac{- x^{3} - 8 x^{2} - 11 x + 20}{x - 1}$

Step 2.3.7.1.5

Divide $- x^{3} - 8 x^{2} - 11 x + 20$ by $x - 1$ .

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Step 2.3.7.1.5.1

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

$x$

$1$

$x^{3}$

$8 x^{2}$

$11 x$

$20$

Step 2.3.7.1.5.2

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

				-	$x^{2}$
	$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$

Step 2.3.7.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			-	$x^{3}$	+	$x^{2}$

Step 2.3.7.1.5.4

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$

Step 2.3.7.1.5.5

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$

Step 2.3.7.1.5.6

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

			-	$x^{2}$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$

Step 2.3.7.1.5.7

Divide the highest order term in the dividend $- 9 x^{2}$ by the highest order term in divisor $x$ .

			-	$x^{2}$	-	$9 x$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$

Step 2.3.7.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$9 x$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$
					-	$9 x^{2}$	+	$9 x$

Step 2.3.7.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{2} + 9 x$

			-	$x^{2}$	-	$9 x$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$
					+	$9 x^{2}$	-	$9 x$

Step 2.3.7.1.5.10

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

			-	$x^{2}$	-	$9 x$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$
					+	$9 x^{2}$	-	$9 x$
							-	$20 x$

Step 2.3.7.1.5.11

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

			-	$x^{2}$	-	$9 x$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$
					+	$9 x^{2}$	-	$9 x$
							-	$20 x$	+	$20$

Step 2.3.7.1.5.12

Divide the highest order term in the dividend $- 20 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	-	$9 x$	-	$20$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$
					+	$9 x^{2}$	-	$9 x$
							-	$20 x$	+	$20$

Step 2.3.7.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$9 x$	-	$20$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$
					+	$9 x^{2}$	-	$9 x$
							-	$20 x$	+	$20$
							-	$20 x$	+	$20$

Step 2.3.7.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x + 20$

			-	$x^{2}$	-	$9 x$	-	$20$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$
					+	$9 x^{2}$	-	$9 x$
							-	$20 x$	+	$20$
							+	$20 x$	-	$20$

Step 2.3.7.1.5.15

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

			-	$x^{2}$	-	$9 x$	-	$20$
$x$	-	$1$	-	$x^{3}$	-	$8 x^{2}$	-	$11 x$	+	$20$
			+	$x^{3}$	-	$x^{2}$
					-	$9 x^{2}$	-	$11 x$
					+	$9 x^{2}$	-	$9 x$
							-	$20 x$	+	$20$
							+	$20 x$	-	$20$
										$0$

Step 2.3.7.1.5.16

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

$- x^{2} - 9 x - 20$

Step 2.3.7.1.6

Write $- x^{3} - 8 x^{2} - 11 x + 20$ as a set of factors.

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{(x - 1) (- x^{2} - 9 x - 20)}$

Step 2.3.7.2

Factor by grouping.

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Step 2.3.7.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 20 = 20$ and whose sum is $b = - 9$ .

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Step 2.3.7.2.1.1

Factor $- 9$ out of $- 9 x$ .

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{(x - 1) (- x^{2} - 9 x - 20)}$

Step 2.3.7.2.1.2

Rewrite $- 9$ as $- 4$ plus $- 5$

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{(x - 1) (- x^{2} + (- 4 - 5) x - 20)}$

Step 2.3.7.2.1.3

Apply the distributive property.

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{(x - 1) (- x^{2} - 4 x - 5 x - 20)}$

Step 2.3.7.2.2

Factor out the greatest common factor from each group.

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Step 2.3.7.2.2.1

Group the first two terms and the last two terms.

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{(x - 1) ((- x^{2} - 4 x) - 5 x - 20)}$

Step 2.3.7.2.2.2

Factor out the greatest common factor (GCF) from each group.

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{(x - 1) (x (- x - 4) + 5 (- x - 4))}$

Step 2.3.7.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 4$ .

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{(x - 1) ((- x - 4) (x + 5))}$

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{(x - 1) (- x - 4) (x + 5)}$

Step 2.4

Cancel the common factor of $- x - 4$ .

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Step 2.4.1

Cancel the common factor.

$f (- x) = \frac{(x + 1) (- x - 4) (x - 5)}{(x - 1) (- x - 4) (x + 5)}$

Step 2.4.2

Rewrite the expression.

$f (- x) = \frac{(x + 1) (x - 5)}{(x - 1) (x + 5)}$

Step 3

A function is even if $f (- x) = f (x)$ .

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Step 3.1

Check if $f (- x) = f (x)$ .

Step 3.2

Since $\frac{(x + 1) (x - 5)}{(x - 1) (x + 5)}$ $\neq$ $\frac{x^{3} - 21 x + 20}{x^{3} - 8 x^{2} + 11 x + 20}$ , the function is not even.

The function is not even

Step 4

A function is odd if $f (- x) = - f (x)$ .

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Step 4.1

Multiply $- 1$ by $\frac{x^{3} - 21 x + 20}{x^{3} - 8 x^{2} + 11 x + 20}$ .

$- f (x) = - \frac{x^{3} - 21 x + 20}{x^{3} - 8 x^{2} + 11 x + 20}$

Step 4.2

Since $\frac{(x + 1) (x - 5)}{(x - 1) (x + 5)}$ $\neq$ $- \frac{x^{3} - 21 x + 20}{x^{3} - 8 x^{2} + 11 x + 20}$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6

Since the function is not odd, it is not symmetric about the origin.

No origin symmetry

Step 7

Since the function is not even, it is not symmetric about the y-axis.

No y-axis symmetry

Step 8

Since the function is neither odd nor even, there is no origin / y-axis symmetry.

Function is not symmetric

Step 9