Find the x and y Intercepts -3x^3+10x^2+32x-35

$- 3 x^{3} + 10 x^{2} + 32 x - 35$

Step 1

Write $- 3 x^{3} + 10 x^{2} + 32 x - 35$ as an equation.

$y = - 3 x^{3} + 10 x^{2} + 32 x - 35$

Step 2

Find the x-intercepts.

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

$0 = - 3 x^{3} + 10 x^{2} + 32 x - 35$

Solve the equation.

Rewrite the equation as $- 3 x^{3} + 10 x^{2} + 32 x - 35 = 0$ .

$- 3 x^{3} + 10 x^{2} + 32 x - 35 = 0$

Factor $- 3 x^{3} + 10 x^{2} + 32 x - 35$ 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.

$p = \pm 1, \pm 35, \pm 5, \pm 7 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 35, \pm 11.\overline{6}, \pm 5, \pm 1.\overline{6}, \pm 7, \pm 2.\overline{3}$

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.

$- 3 \cdot 5^{3} + 10 \cdot 5^{2} + 32 \cdot 5 - 35$

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

$- 3 \cdot 125 + 10 \cdot 5^{2} + 32 \cdot 5 - 35$

Multiply $- 3$ by $125$ .

$- 375 + 10 \cdot 5^{2} + 32 \cdot 5 - 35$

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

$- 375 + 10 \cdot 25 + 32 \cdot 5 - 35$

Multiply $10$ by $25$ .

$- 375 + 250 + 32 \cdot 5 - 35$

Add $- 375$ and $250$ .

$- 125 + 32 \cdot 5 - 35$

Multiply $32$ by $5$ .

$- 125 + 160 - 35$

Add $- 125$ and $160$ .

$35 - 35$

Subtract $35$ from $35$ .

$0$

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.

$\frac{- 3 x^{3} + 10 x^{2} + 32 x - 35}{x - 5}$

Divide $- 3 x^{3} + 10 x^{2} + 32 x - 35$ 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 $0$ .

$x$

$5$

$3 x^{3}$

$10 x^{2}$

$32 x$

$35$

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

				-	$3 x^{2}$
	$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			-	$3 x^{3}$	+	$15 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{3} + 15 x^{2}$

			-	$3 x^{2}$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$

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

			-	$3 x^{2}$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$

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

			-	$3 x^{2}$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

			-	$3 x^{2}$	-	$5 x$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	-	$5 x$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$
					-	$5 x^{2}$	+	$25 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} + 25 x$

			-	$3 x^{2}$	-	$5 x$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$
					+	$5 x^{2}$	-	$25 x$

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

			-	$3 x^{2}$	-	$5 x$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$
					+	$5 x^{2}$	-	$25 x$
							+	$7 x$

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

			-	$3 x^{2}$	-	$5 x$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$
					+	$5 x^{2}$	-	$25 x$
							+	$7 x$	-	$35$

Divide the highest order term in the dividend $7 x$ by the highest order term in divisor $x$ .

			-	$3 x^{2}$	-	$5 x$	+	$7$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$
					+	$5 x^{2}$	-	$25 x$
							+	$7 x$	-	$35$

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	-	$5 x$	+	$7$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$
					+	$5 x^{2}$	-	$25 x$
							+	$7 x$	-	$35$
							+	$7 x$	-	$35$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x - 35$

			-	$3 x^{2}$	-	$5 x$	+	$7$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$
					+	$5 x^{2}$	-	$25 x$
							+	$7 x$	-	$35$
							-	$7 x$	+	$35$

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

			-	$3 x^{2}$	-	$5 x$	+	$7$
$x$	-	$5$	-	$3 x^{3}$	+	$10 x^{2}$	+	$32 x$	-	$35$
			+	$3 x^{3}$	-	$15 x^{2}$
					-	$5 x^{2}$	+	$32 x$
					+	$5 x^{2}$	-	$25 x$
							+	$7 x$	-	$35$
							-	$7 x$	+	$35$
										$0$

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

$- 3 x^{2} - 5 x + 7$

Write $- 3 x^{3} + 10 x^{2} + 32 x - 35$ as a set of factors.

$(x - 5) (- 3 x^{2} - 5 x + 7) = 0$

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

$x - 5 = 0$

$- 3 x^{2} - 5 x + 7 = 0$

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.

$x = 5$

Set $- 3 x^{2} - 5 x + 7$ equal to $0$ and solve for $x$ .

Set $- 3 x^{2} - 5 x + 7$ equal to $0$ .

$- 3 x^{2} - 5 x + 7 = 0$

Solve $- 3 x^{2} - 5 x + 7 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 3$ , $b = - 5$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (- 3 \cdot 7)}}{2 \cdot - 3}$

Simplify.

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 3 \cdot 7}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot 7$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 12 \cdot 7}}{2 \cdot - 3}$

Multiply $12$ by $7$ .

$x = \frac{5 \pm \sqrt{25 + 84}}{2 \cdot - 3}$

Add $25$ and $84$ .

$x = \frac{5 \pm \sqrt{109}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{5 \pm \sqrt{109}}{- 6}$

Move the negative in front of the fraction.

$x = - \frac{5 \pm \sqrt{109}}{6}$

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

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 3 \cdot 7}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot 7$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 12 \cdot 7}}{2 \cdot - 3}$

Multiply $12$ by $7$ .

$x = \frac{5 \pm \sqrt{25 + 84}}{2 \cdot - 3}$

Add $25$ and $84$ .

$x = \frac{5 \pm \sqrt{109}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{5 \pm \sqrt{109}}{- 6}$

Move the negative in front of the fraction.

$x = - \frac{5 \pm \sqrt{109}}{6}$

Change the $\pm$ to $+$ .

$x = - \frac{5 + \sqrt{109}}{6}$

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

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 3 \cdot 7}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot 7$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{5 \pm \sqrt{25 + 12 \cdot 7}}{2 \cdot - 3}$

Multiply $12$ by $7$ .

$x = \frac{5 \pm \sqrt{25 + 84}}{2 \cdot - 3}$

Add $25$ and $84$ .

$x = \frac{5 \pm \sqrt{109}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{5 \pm \sqrt{109}}{- 6}$

Move the negative in front of the fraction.

$x = - \frac{5 \pm \sqrt{109}}{6}$

Change the $\pm$ to $-$ .

$x = - \frac{5 - \sqrt{109}}{6}$

The final answer is the combination of both solutions.

$x = - \frac{5 + \sqrt{109}}{6}, - \frac{5 - \sqrt{109}}{6}$

The final solution is all the values that make $(x - 5) (- 3 x^{2} - 5 x + 7) = 0$ true.

$x = 5, - \frac{5 + \sqrt{109}}{6}, - \frac{5 - \sqrt{109}}{6}$

x-intercept(s) in point form.

x-intercept(s): $(5, 0), (- \frac{5 + \sqrt{109}}{6}, 0), (- \frac{5 - \sqrt{109}}{6}, 0)$

Step 3

Find the y-intercepts.

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

$y = - 3 {(0)}^{3} + 10 {(0)}^{2} + 32 (0) - 35$

Solve the equation.

Remove parentheses.

$y = - 3 \cdot 0^{3} + 10 {(0)}^{2} + 32 (0) - 35$

Remove parentheses.

$y = - 3 \cdot 0^{3} + 10 \cdot 0^{2} + 32 (0) - 35$

Remove parentheses.

$y = - 3 {(0)}^{3} + 10 {(0)}^{2} + 32 (0) - 35$

Simplify $- 3 {(0)}^{3} + 10 {(0)}^{2} + 32 (0) - 35$ .

Simplify each term.

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

$y = - 3 \cdot 0 + 10 {(0)}^{2} + 32 (0) - 35$

Multiply $- 3$ by $0$ .

$y = 0 + 10 {(0)}^{2} + 32 (0) - 35$

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

$y = 0 + 10 \cdot 0 + 32 (0) - 35$

Multiply $10$ by $0$ .

$y = 0 + 0 + 32 (0) - 35$

Multiply $32$ by $0$ .

$y = 0 + 0 + 0 - 35$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 + 0 - 35$

Add $0$ and $0$ .

$y = 0 - 35$

Subtract $35$ from $0$ .

$y = - 35$

y-intercept(s) in point form.

y-intercept(s): $(0, - 35)$

Step 4

List the intersections.

x-intercept(s): $(5, 0), (- \frac{5 + \sqrt{109}}{6}, 0), (- \frac{5 - \sqrt{109}}{6}, 0)$

y-intercept(s): $(0, - 35)$

Step 5

Find the x and y Intercepts -3x^3+10x^2+32x-35

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