Find the x and y Intercepts y=x^4+3x^3-10x^2-24x

$y = x^{4} + 3 x^{3} - 10 x^{2} - 24 x$

Find the x-intercepts.

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

$0 = x^{4} + 3 x^{3} - 10 x^{2} - 24 x$

Solve the equation.

Rewrite the equation as $x^{4} + 3 x^{3} - 10 x^{2} - 24 x = 0$ .

$x^{4} + 3 x^{3} - 10 x^{2} - 24 x = 0$

Factor the left side of the equation.

Factor $x$ out of $x^{4} + 3 x^{3} - 10 x^{2} - 24 x$ .

Factor $x$ out of $x^{4}$ .

$x \cdot x^{3} + 3 x^{3} - 10 x^{2} - 24 x = 0$

Factor $x$ out of $3 x^{3}$ .

$x \cdot x^{3} + x (3 x^{2}) - 10 x^{2} - 24 x = 0$

Factor $x$ out of $- 10 x^{2}$ .

$x \cdot x^{3} + x (3 x^{2}) + x (- 10 x) - 24 x = 0$

Factor $x$ out of $- 24 x$ .

$x \cdot x^{3} + x (3 x^{2}) + x (- 10 x) + x \cdot - 24 = 0$

Factor $x$ out of $x \cdot x^{3} + x (3 x^{2})$ .

$x (x^{3} + 3 x^{2}) + x (- 10 x) + x \cdot - 24 = 0$

Factor $x$ out of $x (x^{3} + 3 x^{2}) + x (- 10 x)$ .

$x (x^{3} + 3 x^{2} - 10 x) + x \cdot - 24 = 0$

Factor $x$ out of $x (x^{3} + 3 x^{2} - 10 x) + x \cdot - 24$ .

$x (x^{3} + 3 x^{2} - 10 x - 24) = 0$

Factor $x^{3} + 3 x^{2} - 10 x - 24$ 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.

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

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

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

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.

${(- 2)}^{3} + 3 {(- 2)}^{2} - 10 \cdot - 2 - 24$

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

$- 8 + 3 {(- 2)}^{2} - 10 \cdot - 2 - 24$

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

$- 8 + 3 \cdot 4 - 10 \cdot - 2 - 24$

Multiply $3$ by $4$ .

$- 8 + 12 - 10 \cdot - 2 - 24$

Add $- 8$ and $12$ .

$4 - 10 \cdot - 2 - 24$

Multiply $- 10$ by $- 2$ .

$4 + 20 - 24$

Add $4$ and $20$ .

$24 - 24$

Subtract $24$ from $24$ .

$0$

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.

$\frac{x^{3} + 3 x^{2} - 10 x - 24}{x + 2}$

Divide $x^{3} + 3 x^{2} - 10 x - 24$ 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 $0$ .

$x$

$2$

$x^{3}$

$3 x^{2}$

$10 x$

$24$

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			+	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$

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

				$x^{2}$	+	$x$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$
					+	$x^{2}$	+	$2 x$

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

				$x^{2}$	+	$x$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$
					-	$x^{2}$	-	$2 x$

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

				$x^{2}$	+	$x$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$
					-	$x^{2}$	-	$2 x$
							-	$12 x$

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

				$x^{2}$	+	$x$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$
					-	$x^{2}$	-	$2 x$
							-	$12 x$	-	$24$

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$x$	-	$12$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$
					-	$x^{2}$	-	$2 x$
							-	$12 x$	-	$24$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	-	$12$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$
					-	$x^{2}$	-	$2 x$
							-	$12 x$	-	$24$
							-	$12 x$	-	$24$

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x - 24$

				$x^{2}$	+	$x$	-	$12$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$
					-	$x^{2}$	-	$2 x$
							-	$12 x$	-	$24$
							+	$12 x$	+	$24$

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

				$x^{2}$	+	$x$	-	$12$
$x$	+	$2$		$x^{3}$	+	$3 x^{2}$	-	$10 x$	-	$24$
			-	$x^{3}$	-	$2 x^{2}$
					+	$x^{2}$	-	$10 x$
					-	$x^{2}$	-	$2 x$
							-	$12 x$	-	$24$
							+	$12 x$	+	$24$
										$0$

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

$x^{2} + x - 12$

Write $x^{3} + 3 x^{2} - 10 x - 24$ as a set of factors.

$x ((x + 2) (x^{2} + x - 12)) = 0$

Factor.

Factor $x^{2} + x - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

$x ((x + 2) ((x - 3) (x + 4))) = 0$

Remove unnecessary parentheses.

$x ((x + 2) (x - 3) (x + 4)) = 0$

Remove unnecessary parentheses.

$x (x + 2) (x - 3) (x + 4) = 0$

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

$x = 0$

$x + 2 = 0$

$x - 3 = 0$

$x + 4 = 0$

Set $x$ equal to $0$ .

$x = 0$

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 $x - 3$ equal to $0$ and solve for $x$ .

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

$x = 0, - 2, 3, - 4$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 2, 0), (3, 0), (- 4, 0)$

Find the y-intercepts.

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

$y = {(0)}^{4} + 3 {(0)}^{3} - 10 {(0)}^{2} - 24 \cdot 0$

Solve the equation.

Remove parentheses.

$y = 0^{4} + 3 {(0)}^{3} - 10 {(0)}^{2} - 24 \cdot 0$

Remove parentheses.

$y = 0^{4} + 3 \cdot 0^{3} - 10 {(0)}^{2} - 24 \cdot 0$

Remove parentheses.

$y = 0^{4} + 3 \cdot 0^{3} - 10 \cdot 0^{2} - 24 \cdot 0$

Remove parentheses.

$y = {(0)}^{4} + 3 {(0)}^{3} - 10 {(0)}^{2} - 24 \cdot 0$

Simplify ${(0)}^{4} + 3 {(0)}^{3} - 10 {(0)}^{2} - 24 \cdot 0$ .

Simplify each term.

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

$y = 0 + 3 {(0)}^{3} - 10 {(0)}^{2} - 24 \cdot 0$

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

$y = 0 + 3 \cdot 0 - 10 {(0)}^{2} - 24 \cdot 0$

Multiply $3$ by $0$ .

$y = 0 + 0 - 10 {(0)}^{2} - 24 \cdot 0$

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

$y = 0 + 0 - 10 \cdot 0 - 24 \cdot 0$

Multiply $- 10$ by $0$ .

$y = 0 + 0 + 0 - 24 \cdot 0$

Multiply $- 24$ by $0$ .

$y = 0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 0$

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

List the intersections.

x-intercept(s): $(0, 0), (- 2, 0), (3, 0), (- 4, 0)$

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

Find the x and y Intercepts y=x^4+3x^3-10x^2-24x

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