Find the x and y Intercepts x^3-x^2-16x-20

$x^{3} - x^{2} - 16 x - 20$

Step 1

Write $x^{3} - x^{2} - 16 x - 20$ as an equation.

$y = x^{3} - x^{2} - 16 x - 20$

Step 2

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = x^{3} - x^{2} - 16 x - 20$

Solve the equation.

Rewrite the equation as $x^{3} - x^{2} - 16 x - 20 = 0$ .

$x^{3} - x^{2} - 16 x - 20 = 0$

Factor the left side of the equation.

Factor $x^{3} - x^{2} - 16 x - 20$ using the rational roots test.

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.

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

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

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

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

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - {(- 2)}^{2} - 16 \cdot - 2 - 20$

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

$- 8 - {(- 2)}^{2} - 16 \cdot - 2 - 20$

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

$- 8 - 1 \cdot 4 - 16 \cdot - 2 - 20$

Multiply $- 1$ by $4$ .

$- 8 - 4 - 16 \cdot - 2 - 20$

Subtract $4$ from $- 8$ .

$- 12 - 16 \cdot - 2 - 20$

Multiply $- 16$ by $- 2$ .

$- 12 + 32 - 20$

Add $- 12$ and $32$ .

$20 - 20$

Subtract $20$ from $20$ .

$0$

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.

$\frac{x^{3} - x^{2} - 16 x - 20}{x + 2}$

Divide $x^{3} - x^{2} - 16 x - 20$ by $x + 2$ .

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

$x$

$2$

$x^{3}$

$x^{2}$

$16 x$

$20$

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			+	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$

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

				$x^{2}$	-	$3 x$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$
					-	$3 x^{2}$	-	$6 x$

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

				$x^{2}$	-	$3 x$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$
					+	$3 x^{2}$	+	$6 x$

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

				$x^{2}$	-	$3 x$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$
					+	$3 x^{2}$	+	$6 x$
							-	$10 x$

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

				$x^{2}$	-	$3 x$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$
					+	$3 x^{2}$	+	$6 x$
							-	$10 x$	-	$20$

Divide the highest order term in the dividend $- 10 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$3 x$	-	$10$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$
					+	$3 x^{2}$	+	$6 x$
							-	$10 x$	-	$20$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$	-	$10$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$
					+	$3 x^{2}$	+	$6 x$
							-	$10 x$	-	$20$
							-	$10 x$	-	$20$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x - 20$

				$x^{2}$	-	$3 x$	-	$10$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$
					+	$3 x^{2}$	+	$6 x$
							-	$10 x$	-	$20$
							+	$10 x$	+	$20$

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

				$x^{2}$	-	$3 x$	-	$10$
$x$	+	$2$		$x^{3}$	-	$x^{2}$	-	$16 x$	-	$20$
			-	$x^{3}$	-	$2 x^{2}$
					-	$3 x^{2}$	-	$16 x$
					+	$3 x^{2}$	+	$6 x$
							-	$10 x$	-	$20$
							+	$10 x$	+	$20$
										$0$

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

$x^{2} - 3 x - 10$

Write $x^{3} - x^{2} - 16 x - 20$ as a set of factors.

$(x + 2) (x^{2} - 3 x - 10) = 0$

Factor $x^{2} - 3 x - 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

$(x + 2) ((x - 5) (x + 2)) = 0$

Remove unnecessary parentheses.

$(x + 2) (x - 5) (x + 2) = 0$

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

$x + 2 = 0$

$x - 5 = 0$

$x + 2 = 0$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(x + 2) (x - 5) (x + 2) = 0$ true.

$x = - 2, 5$

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0), (5, 0)$

Step 3

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = {(0)}^{3} - {(0)}^{2} - 16 \cdot 0 - 20$

Solve the equation.

Remove parentheses.

$y = 0^{3} - {(0)}^{2} - 16 \cdot 0 - 20$

Remove parentheses.

$y = 0^{3} - 0^{2} - 16 \cdot 0 - 20$

Remove parentheses.

$y = {(0)}^{3} - {(0)}^{2} - 16 \cdot 0 - 20$

Simplify ${(0)}^{3} - {(0)}^{2} - 16 \cdot 0 - 20$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$y = 0 - {(0)}^{2} - 16 \cdot 0 - 20$

Raising $0$ to any positive power yields $0$ .

$y = 0 - 0 - 16 \cdot 0 - 20$

Multiply $- 1$ by $0$ .

$y = 0 + 0 - 16 \cdot 0 - 20$

Multiply $- 16$ by $0$ .

$y = 0 + 0 + 0 - 20$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 + 0 - 20$

Add $0$ and $0$ .

$y = 0 - 20$

Subtract $20$ from $0$ .

$y = - 20$

y-intercept(s) in point form.

y-intercept(s): $(0, - 20)$

Step 4

List the intersections.

x-intercept(s): $(- 2, 0), (5, 0)$

y-intercept(s): $(0, - 20)$

Step 5

Find the x and y Intercepts x^3-x^2-16x-20

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