Find the x and y Intercepts F(x)=x^3-5x^2-8x+48

$F (x) = x^{3} - 5 x^{2} - 8 x + 48$

Step 1

Find the x-intercepts.

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

$0 = x^{3} - 5 x^{2} - 8 x + 48$

Solve the equation.

Rewrite the equation as $x^{3} - 5 x^{2} - 8 x + 48 = 0$ .

$x^{3} - 5 x^{2} - 8 x + 48 = 0$

Factor the left side of the equation.

Factor $x^{3} - 5 x^{2} - 8 x + 48$ 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.

$p = \pm 1, \pm 48, \pm 2, \pm 24, \pm 3, \pm 16, \pm 4, \pm 12, \pm 6, \pm 8 q = \pm 1$

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

$\pm 1, \pm 48, \pm 2, \pm 24, \pm 3, \pm 16, \pm 4, \pm 12, \pm 6, \pm 8$

Substitute $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

Substitute $- 3$ into the polynomial.

${(- 3)}^{3} - 5 {(- 3)}^{2} - 8 \cdot - 3 + 48$

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

$- 27 - 5 {(- 3)}^{2} - 8 \cdot - 3 + 48$

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

$- 27 - 5 \cdot 9 - 8 \cdot - 3 + 48$

Multiply $- 5$ by $9$ .

$- 27 - 45 - 8 \cdot - 3 + 48$

Subtract $45$ from $- 27$ .

$- 72 - 8 \cdot - 3 + 48$

Multiply $- 8$ by $- 3$ .

$- 72 + 24 + 48$

Add $- 72$ and $24$ .

$- 48 + 48$

Add $- 48$ and $48$ .

$0$

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 5 x^{2} - 8 x + 48}{x + 3}$

Divide $x^{3} - 5 x^{2} - 8 x + 48$ by $x + 3$ .

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

$x$

$3$

$x^{3}$

$5 x^{2}$

$8 x$

$48$

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			+	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					-	$8 x^{2}$	-	$24 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} - 24 x$

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$

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

				$x^{2}$	-	$8 x$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$
							+	$16 x$	+	$48$

The expression needs to be subtracted from the dividend, so change all the signs in $16 x + 48$

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$
							-	$16 x$	-	$48$

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

				$x^{2}$	-	$8 x$	+	$16$
$x$	+	$3$		$x^{3}$	-	$5 x^{2}$	-	$8 x$	+	$48$
			-	$x^{3}$	-	$3 x^{2}$
					-	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	+	$24 x$
							+	$16 x$	+	$48$
							-	$16 x$	-	$48$
										$0$

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

$x^{2} - 8 x + 16$

Write $x^{3} - 5 x^{2} - 8 x + 48$ as a set of factors.

$(x + 3) (x^{2} - 8 x + 16) = 0$

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$(x + 3) (x^{2} - 8 x + 4^{2}) = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$(x + 3) (x^{2} - 2 \cdot x \cdot 4 + 4^{2}) = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

$(x + 3) {(x - 4)}^{2} = 0$

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

$x + 3 = 0$

${(x - 4)}^{2} = 0$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Set ${(x - 4)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 4)}^{2}$ equal to $0$ .

${(x - 4)}^{2} = 0$

Solve ${(x - 4)}^{2} = 0$ for $x$ .

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(x + 3) {(x - 4)}^{2} = 0$ true.

$x = - 3, 4$

x-intercept(s) in point form.

x-intercept(s): $(- 3, 0), (4, 0)$

Step 2

Find the y-intercepts.

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

$y = {(0)}^{3} - 5 {(0)}^{2} - 8 \cdot 0 + 48$

Solve the equation.

Remove parentheses.

$y = 0^{3} - 5 {(0)}^{2} - 8 \cdot 0 + 48$

Remove parentheses.

$y = 0^{3} - 5 \cdot 0^{2} - 8 \cdot 0 + 48$

Remove parentheses.

$y = {(0)}^{3} - 5 {(0)}^{2} - 8 \cdot 0 + 48$

Simplify ${(0)}^{3} - 5 {(0)}^{2} - 8 \cdot 0 + 48$ .

Simplify each term.

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

$y = 0 - 5 {(0)}^{2} - 8 \cdot 0 + 48$

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

$y = 0 - 5 \cdot 0 - 8 \cdot 0 + 48$

Multiply $- 5$ by $0$ .

$y = 0 + 0 - 8 \cdot 0 + 48$

Multiply $- 8$ by $0$ .

$y = 0 + 0 + 0 + 48$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 48$

Add $0$ and $0$ .

$y = 0 + 48$

Add $0$ and $48$ .

$y = 48$

y-intercept(s) in point form.

y-intercept(s): $(0, 48)$

Step 3

List the intersections.

x-intercept(s): $(- 3, 0), (4, 0)$

y-intercept(s): $(0, 48)$

Step 4

Find the x and y Intercepts F(x)=x^3-5x^2-8x+48

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