Find the Vertical and Horizontal Intercept(s) f(x)=(2x^3+24x^2+35x-175)/(x^2+14x+45)

$f (x) = \frac{2 x^{3} + 24 x^{2} + 35 x - 175}{x^{2} + 14 x + 45}$

Step 1

Find the x-intercepts.

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

$0 = \frac{2 x^{3} + 24 x^{2} + 35 x - 175}{x^{2} + 14 x + 45}$

Solve the equation.

Set the numerator equal to zero.

$2 x^{3} + 24 x^{2} + 35 x - 175 = 0$

Solve the equation for $x$ .

Factor $2 x^{3} + 24 x^{2} + 35 x - 175$ 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.

$p = \pm 1, \pm 175, \pm 5, \pm 35, \pm 7, \pm 25 q = \pm 1, \pm 2$

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

$\pm 1, \pm 0.5, \pm 175, \pm 87.5, \pm 5, \pm 2.5, \pm 35, \pm 17.5, \pm 7, \pm 3.5, \pm 25, \pm 12.5$

Substitute $- 5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 5$ is a root of the polynomial.

Substitute $- 5$ into the polynomial.

$2 {(- 5)}^{3} + 24 {(- 5)}^{2} + 35 \cdot - 5 - 175$

Raise $- 5$ to the power of $3$ .

$2 \cdot - 125 + 24 {(- 5)}^{2} + 35 \cdot - 5 - 175$

Multiply $2$ by $- 125$ .

$- 250 + 24 {(- 5)}^{2} + 35 \cdot - 5 - 175$

Raise $- 5$ to the power of $2$ .

$- 250 + 24 \cdot 25 + 35 \cdot - 5 - 175$

Multiply $24$ by $25$ .

$- 250 + 600 + 35 \cdot - 5 - 175$

Add $- 250$ and $600$ .

$350 + 35 \cdot - 5 - 175$

Multiply $35$ by $- 5$ .

$350 - 175 - 175$

Subtract $175$ from $350$ .

$175 - 175$

Subtract $175$ from $175$ .

$0$

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} + 24 x^{2} + 35 x - 175}{x + 5}$

Divide $2 x^{3} + 24 x^{2} + 35 x - 175$ by $x + 5$ .

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

$x$

$5$

$2 x^{3}$

$24 x^{2}$

$35 x$

$175$

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

					$2 x^{2}$
	$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			+	$2 x^{3}$	+	$10 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 10 x^{2}$

				$2 x^{2}$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$

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

				$2 x^{2}$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$

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

				$2 x^{2}$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$

Divide the highest order term in the dividend $14 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$14 x$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$14 x$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$
					+	$14 x^{2}$	+	$70 x$

The expression needs to be subtracted from the dividend, so change all the signs in $14 x^{2} + 70 x$

				$2 x^{2}$	+	$14 x$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$
					-	$14 x^{2}$	-	$70 x$

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

				$2 x^{2}$	+	$14 x$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$
					-	$14 x^{2}$	-	$70 x$
							-	$35 x$

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

				$2 x^{2}$	+	$14 x$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$
					-	$14 x^{2}$	-	$70 x$
							-	$35 x$	-	$175$

Divide the highest order term in the dividend $- 35 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$14 x$	-	$35$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$
					-	$14 x^{2}$	-	$70 x$
							-	$35 x$	-	$175$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$14 x$	-	$35$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$
					-	$14 x^{2}$	-	$70 x$
							-	$35 x$	-	$175$
							-	$35 x$	-	$175$

The expression needs to be subtracted from the dividend, so change all the signs in $- 35 x - 175$

				$2 x^{2}$	+	$14 x$	-	$35$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$
					-	$14 x^{2}$	-	$70 x$
							-	$35 x$	-	$175$
							+	$35 x$	+	$175$

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

				$2 x^{2}$	+	$14 x$	-	$35$
$x$	+	$5$		$2 x^{3}$	+	$24 x^{2}$	+	$35 x$	-	$175$
			-	$2 x^{3}$	-	$10 x^{2}$
					+	$14 x^{2}$	+	$35 x$
					-	$14 x^{2}$	-	$70 x$
							-	$35 x$	-	$175$
							+	$35 x$	+	$175$
										$0$

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

$2 x^{2} + 14 x - 35$

Write $2 x^{3} + 24 x^{2} + 35 x - 175$ as a set of factors.

$(x + 5) (2 x^{2} + 14 x - 35) = 0$

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

$x + 5 = 0$

$2 x^{2} + 14 x - 35 = 0$

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

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

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Set $2 x^{2} + 14 x - 35$ equal to $0$ and solve for $x$ .

Set $2 x^{2} + 14 x - 35$ equal to $0$ .

$2 x^{2} + 14 x - 35 = 0$

Solve $2 x^{2} + 14 x - 35 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 2$ , $b = 14$ , and $c = - 35$ into the quadratic formula and solve for $x$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (2 \cdot - 35)}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 2 \cdot - 35}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 35$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 8 \cdot - 35}}{2 \cdot 2}$

Multiply $- 8$ by $- 35$ .

$x = \frac{- 14 \pm \sqrt{196 + 280}}{2 \cdot 2}$

Add $196$ and $280$ .

$x = \frac{- 14 \pm \sqrt{476}}{2 \cdot 2}$

Rewrite $476$ as $2^{2} \cdot 119$ .

Factor $4$ out of $476$ .

$x = \frac{- 14 \pm \sqrt{4 (119)}}{2 \cdot 2}$

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

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 119}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{119}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 14 \pm 2 \sqrt{119}}{4}$

Simplify $\frac{- 14 \pm 2 \sqrt{119}}{4}$ .

$x = \frac{- 7 \pm \sqrt{119}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 2 \cdot - 35}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 35$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 8 \cdot - 35}}{2 \cdot 2}$

Multiply $- 8$ by $- 35$ .

$x = \frac{- 14 \pm \sqrt{196 + 280}}{2 \cdot 2}$

Add $196$ and $280$ .

$x = \frac{- 14 \pm \sqrt{476}}{2 \cdot 2}$

Rewrite $476$ as $2^{2} \cdot 119$ .

Factor $4$ out of $476$ .

$x = \frac{- 14 \pm \sqrt{4 (119)}}{2 \cdot 2}$

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

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 119}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{119}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 14 \pm 2 \sqrt{119}}{4}$

Simplify $\frac{- 14 \pm 2 \sqrt{119}}{4}$ .

$x = \frac{- 7 \pm \sqrt{119}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 7 + \sqrt{119}}{2}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{119}}{2}$

Factor $- 1$ out of $\sqrt{119}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{119})}{2}$

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{119})$ .

$x = \frac{- 1 (7 - \sqrt{119})}{2}$

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{119}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 2 \cdot - 35}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 35$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 8 \cdot - 35}}{2 \cdot 2}$

Multiply $- 8$ by $- 35$ .

$x = \frac{- 14 \pm \sqrt{196 + 280}}{2 \cdot 2}$

Add $196$ and $280$ .

$x = \frac{- 14 \pm \sqrt{476}}{2 \cdot 2}$

Rewrite $476$ as $2^{2} \cdot 119$ .

Factor $4$ out of $476$ .

$x = \frac{- 14 \pm \sqrt{4 (119)}}{2 \cdot 2}$

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

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 119}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{119}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 14 \pm 2 \sqrt{119}}{4}$

Simplify $\frac{- 14 \pm 2 \sqrt{119}}{4}$ .

$x = \frac{- 7 \pm \sqrt{119}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 7 - \sqrt{119}}{2}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{119}}{2}$

Factor $- 1$ out of $- \sqrt{119}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{119})}{2}$

Factor $- 1$ out of $- 1 (7) - (\sqrt{119})$ .

$x = \frac{- 1 (7 + \sqrt{119})}{2}$

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{119}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{119}}{2}, - \frac{7 + \sqrt{119}}{2}$

The final solution is all the values that make $(x + 5) (2 x^{2} + 14 x - 35) = 0$ true.

$x = - 5, - \frac{7 - \sqrt{119}}{2}, - \frac{7 + \sqrt{119}}{2}$

Exclude the solutions that do not make $0 = \frac{2 x^{3} + 24 x^{2} + 35 x - 175}{x^{2} + 14 x + 45}$ true.

$x = - \frac{7 - \sqrt{119}}{2}, - \frac{7 + \sqrt{119}}{2}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{7 - \sqrt{119}}{2}, 0), (- \frac{7 + \sqrt{119}}{2}, 0)$

Step 2

Find the y-intercepts.

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

$y = \frac{2 {(0)}^{3} + 24 {(0)}^{2} + 35 (0) - 175}{{(0)}^{2} + 14 (0) + 45}$

Solve the equation.

Remove parentheses.

$y = \frac{2 \cdot 0^{3} + 24 {(0)}^{2} + 35 (0) - 175}{{(0)}^{2} + 14 (0) + 45}$

Remove parentheses.

$y = \frac{2 \cdot 0^{3} + 24 \cdot 0^{2} + 35 (0) - 175}{{(0)}^{2} + 14 (0) + 45}$

Remove parentheses.

$y = \frac{2 \cdot 0^{3} + 24 \cdot 0^{2} + 35 (0) - 175}{0^{2} + 14 (0) + 45}$

Remove parentheses.

$y = \frac{2 {(0)}^{3} + 24 {(0)}^{2} + 35 (0) - 175}{{(0)}^{2} + 14 (0) + 45}$

Simplify $\frac{2 {(0)}^{3} + 24 {(0)}^{2} + 35 (0) - 175}{{(0)}^{2} + 14 (0) + 45}$ .

Simplify the numerator.

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

$y = \frac{2 \cdot 0 + 24 \cdot 0^{2} + 35 (0) - 175}{0^{2} + 14 (0) + 45}$

Multiply $2$ by $0$ .

$y = \frac{0 + 24 \cdot 0^{2} + 35 (0) - 175}{0^{2} + 14 (0) + 45}$

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

$y = \frac{0 + 24 \cdot 0 + 35 (0) - 175}{0^{2} + 14 (0) + 45}$

Multiply $24$ by $0$ .

$y = \frac{0 + 0 + 35 (0) - 175}{0^{2} + 14 (0) + 45}$

Multiply $35$ by $0$ .

$y = \frac{0 + 0 + 0 - 175}{0^{2} + 14 (0) + 45}$

Add $0$ and $0$ .

$y = \frac{0 + 0 - 175}{0^{2} + 14 (0) + 45}$

Add $0$ and $0$ .

$y = \frac{0 - 175}{0^{2} + 14 (0) + 45}$

Subtract $175$ from $0$ .

$y = \frac{- 175}{0^{2} + 14 (0) + 45}$

Simplify the denominator.

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

$y = \frac{- 175}{0 + 14 (0) + 45}$

Multiply $14$ by $0$ .

$y = \frac{- 175}{0 + 0 + 45}$

Add $0$ and $0$ .

$y = \frac{- 175}{0 + 45}$

Add $0$ and $45$ .

$y = \frac{- 175}{45}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 175$ and $45$ .

Factor $5$ out of $- 175$ .

$y = \frac{5 (- 35)}{45}$

Cancel the common factors.

Factor $5$ out of $45$ .

$y = \frac{5 \cdot - 35}{5 \cdot 9}$

Cancel the common factor.

$y = \frac{5 \cdot - 35}{5 \cdot 9}$

Rewrite the expression.

$y = \frac{- 35}{9}$

Move the negative in front of the fraction.

$y = - \frac{35}{9}$

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{35}{9})$

Step 3

List the intersections.

x-intercept(s): $(- \frac{7 - \sqrt{119}}{2}, 0), (- \frac{7 + \sqrt{119}}{2}, 0)$

y-intercept(s): $(0, - \frac{35}{9})$

Step 4

Find the Vertical and Horizontal Intercept(s) f(x)=(2x^3+24x^2+35x-175)/(x^2+14x+45)

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