Identify the Zeros and Their Multiplicities 2x^4-7x^3-17x^2+58x-24

$2 x^{4} - 7 x^{3} - 17 x^{2} + 58 x - 24$

Step 1

Set $2 x^{4} - 7 x^{3} - 17 x^{2} + 58 x - 24$ equal to $0$ .

$2 x^{4} - 7 x^{3} - 17 x^{2} + 58 x - 24 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $2 x^{4} - 7 x^{3} - 17 x^{2} + 58 x - 24$ 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.

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

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

$\pm 1, \pm 0.5, \pm 24, \pm 12, \pm 2, \pm 6, \pm 3, \pm 1.5, \pm 8, \pm 4$

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

Substitute $0.5$ into the polynomial.

$2 \cdot {0.5}^{4} - 7 \cdot {0.5}^{3} - 17 \cdot {0.5}^{2} + 58 \cdot 0.5 - 24$

Raise $0.5$ to the power of $4$ .

$2 \cdot 0.0625 - 7 \cdot {0.5}^{3} - 17 \cdot {0.5}^{2} + 58 \cdot 0.5 - 24$

Multiply $2$ by $0.0625$ .

$0.125 - 7 \cdot {0.5}^{3} - 17 \cdot {0.5}^{2} + 58 \cdot 0.5 - 24$

Raise $0.5$ to the power of $3$ .

$0.125 - 7 \cdot 0.125 - 17 \cdot {0.5}^{2} + 58 \cdot 0.5 - 24$

Multiply $- 7$ by $0.125$ .

$0.125 - 0.875 - 17 \cdot {0.5}^{2} + 58 \cdot 0.5 - 24$

Subtract $0.875$ from $0.125$ .

$- 0.75 - 17 \cdot {0.5}^{2} + 58 \cdot 0.5 - 24$

Raise $0.5$ to the power of $2$ .

$- 0.75 - 17 \cdot 0.25 + 58 \cdot 0.5 - 24$

Multiply $- 17$ by $0.25$ .

$- 0.75 - 4.25 + 58 \cdot 0.5 - 24$

Subtract $4.25$ from $- 0.75$ .

$- 5 + 58 \cdot 0.5 - 24$

Multiply $58$ by $0.5$ .

$- 5 + 29 - 24$

Add $- 5$ and $29$ .

$24 - 24$

Subtract $24$ from $24$ .

$0$

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{4} - 7 x^{3} - 17 x^{2} + 58 x - 24}{2 x - 1}$

Divide $2 x^{4} - 7 x^{3} - 17 x^{2} + 58 x - 24$ by $2 x - 1$ .

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

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$17 x^{2}$

$58 x$

$24$

Divide the highest order term in the dividend $2 x^{4}$ by the highest order term in divisor $2 x$ .

					$x^{3}$
	$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$

Multiply the new quotient term by the divisor.

				$x^{3}$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			+	$2 x^{4}$	-	$x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} - x^{3}$

				$x^{3}$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$

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

				$x^{3}$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$
					-	$6 x^{3}$

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

				$x^{3}$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$
					-	$6 x^{3}$	-	$17 x^{2}$

Divide the highest order term in the dividend $- 6 x^{3}$ by the highest order term in divisor $2 x$ .

				$x^{3}$	-	$3 x^{2}$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$
					-	$6 x^{3}$	-	$17 x^{2}$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$3 x^{2}$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$
					-	$6 x^{3}$	-	$17 x^{2}$
					-	$6 x^{3}$	+	$3 x^{2}$

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

				$x^{3}$	-	$3 x^{2}$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$
					-	$6 x^{3}$	-	$17 x^{2}$
					+	$6 x^{3}$	-	$3 x^{2}$

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

				$x^{3}$	-	$3 x^{2}$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$
					-	$6 x^{3}$	-	$17 x^{2}$
					+	$6 x^{3}$	-	$3 x^{2}$
							-	$20 x^{2}$

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

$x^{3}$

$3 x^{2}$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$17 x^{2}$

$58 x$

$24$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$17 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x^{2}$

$58 x$

Divide the highest order term in the dividend $- 20 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$3 x^{2}$

$10 x$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$17 x^{2}$

$58 x$

$24$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$17 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x^{2}$

$58 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$10 x$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$17 x^{2}$

$58 x$

$24$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$17 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x^{2}$

$58 x$

$20 x^{2}$

$10 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x^{2} + 10 x$

$x^{3}$

$3 x^{2}$

$10 x$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$17 x^{2}$

$58 x$

$24$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$17 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x^{2}$

$58 x$

$20 x^{2}$

$10 x$

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

$x^{3}$

$3 x^{2}$

$10 x$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$17 x^{2}$

$58 x$

$24$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$17 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x^{2}$

$58 x$

$20 x^{2}$

$10 x$

$48 x$

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

$x^{3}$

$3 x^{2}$

$10 x$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$17 x^{2}$

$58 x$

$24$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$17 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x^{2}$

$58 x$

$20 x^{2}$

$10 x$

$48 x$

$24$

Divide the highest order term in the dividend $48 x$ by the highest order term in divisor $2 x$ .

$x^{3}$

$3 x^{2}$

$10 x$

$24$

$2 x$

$1$

$2 x^{4}$

$7 x^{3}$

$17 x^{2}$

$58 x$

$24$

$2 x^{4}$

$x^{3}$

$6 x^{3}$

$17 x^{2}$

$6 x^{3}$

$3 x^{2}$

$20 x^{2}$

$58 x$

$20 x^{2}$

$10 x$

$48 x$

$24$

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$3 x^{2}$	-	$10 x$	+	$24$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$
					-	$6 x^{3}$	-	$17 x^{2}$
					+	$6 x^{3}$	-	$3 x^{2}$
							-	$20 x^{2}$	+	$58 x$
							+	$20 x^{2}$	-	$10 x$
									+	$48 x$	-	$24$
									+	$48 x$	-	$24$

The expression needs to be subtracted from the dividend, so change all the signs in $48 x - 24$

				$x^{3}$	-	$3 x^{2}$	-	$10 x$	+	$24$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$
					-	$6 x^{3}$	-	$17 x^{2}$
					+	$6 x^{3}$	-	$3 x^{2}$
							-	$20 x^{2}$	+	$58 x$
							+	$20 x^{2}$	-	$10 x$
									+	$48 x$	-	$24$
									-	$48 x$	+	$24$

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

				$x^{3}$	-	$3 x^{2}$	-	$10 x$	+	$24$
$2 x$	-	$1$		$2 x^{4}$	-	$7 x^{3}$	-	$17 x^{2}$	+	$58 x$	-	$24$
			-	$2 x^{4}$	+	$x^{3}$
					-	$6 x^{3}$	-	$17 x^{2}$
					+	$6 x^{3}$	-	$3 x^{2}$
							-	$20 x^{2}$	+	$58 x$
							+	$20 x^{2}$	-	$10 x$
									+	$48 x$	-	$24$
									-	$48 x$	+	$24$
												$0$

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

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

Write $2 x^{4} - 7 x^{3} - 17 x^{2} + 58 x - 24$ as a set of factors.

$(2 x - 1) (x^{3} - 3 x^{2} - 10 x + 24) = 0$

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.

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

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

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

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

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

Multiply $- 3$ by $4$ .

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

Subtract $12$ from $8$ .

$- 4 - 10 \cdot 2 + 24$

Multiply $- 10$ by $2$ .

$- 4 - 20 + 24$

Subtract $20$ from $- 4$ .

$- 24 + 24$

Add $- 24$ and $24$ .

$0$

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.

$(2 x - 1) ((x - 2) (x^{2} - x - 12)) = 0$

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 $- 1$ .

$- 4, 3$

Write the factored form using these integers.

$(2 x - 1) ((x - 2) ((x - 4) (x + 3))) = 0$

Remove unnecessary parentheses.

$(2 x - 1) ((x - 2) (x - 4) (x + 3)) = 0$

Remove unnecessary parentheses.

$(2 x - 1) (x - 2) (x - 4) (x + 3) = 0$

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

$2 x - 1 = 0$

$x - 2 = 0$

$x - 4 = 0$

$x + 3 = 0$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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.

$x = - 3$

The final solution is all the values that make $(2 x - 1) (x - 2) (x - 4) (x + 3) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = \frac{1}{2}$ (Multiplicity of $1$ )

$x = 2$ (Multiplicity of $1$ )

$x = 4$ (Multiplicity of $1$ )

$x = - 3$ (Multiplicity of $1$ )

$x = \frac{1}{2}$ (Multiplicity of $1$ )

$x = 2$ (Multiplicity of $1$ )

$x = 4$ (Multiplicity of $1$ )

$x = - 3$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities 2x^4-7x^3-17x^2+58x-24

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