Identify the Zeros and Their Multiplicities 4x^4-12x^3+x^2+12x+4

$4 x^{4} - 12 x^{3} + x^{2} + 12 x + 4$

Step 1

Set $4 x^{4} - 12 x^{3} + x^{2} + 12 x + 4$ equal to $0$ .

$4 x^{4} - 12 x^{3} + x^{2} + 12 x + 4 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $4 x^{4} - 12 x^{3} + x^{2} + 12 x + 4$ 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.

$p = \pm 1, \pm 4, \pm 2 q = \pm 1, \pm 4, \pm 2$

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 4, \pm 2$

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.

$4 {(- 0.5)}^{4} - 12 {(- 0.5)}^{3} + {(- 0.5)}^{2} + 12 \cdot - 0.5 + 4$

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

$4 \cdot 0.0625 - 12 {(- 0.5)}^{3} + {(- 0.5)}^{2} + 12 \cdot - 0.5 + 4$

Multiply $4$ by $0.0625$ .

$0.25 - 12 {(- 0.5)}^{3} + {(- 0.5)}^{2} + 12 \cdot - 0.5 + 4$

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

$0.25 - 12 \cdot - 0.125 + {(- 0.5)}^{2} + 12 \cdot - 0.5 + 4$

Multiply $- 12$ by $- 0.125$ .

$0.25 + 1.5 + {(- 0.5)}^{2} + 12 \cdot - 0.5 + 4$

Add $0.25$ and $1.5$ .

$1.75 + {(- 0.5)}^{2} + 12 \cdot - 0.5 + 4$

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

$1.75 + 0.25 + 12 \cdot - 0.5 + 4$

Add $1.75$ and $0.25$ .

$2 + 12 \cdot - 0.5 + 4$

Multiply $12$ by $- 0.5$ .

$2 - 6 + 4$

Subtract $6$ from $2$ .

$- 4 + 4$

Add $- 4$ and $4$ .

$0$

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.

$\frac{4 x^{4} - 12 x^{3} + x^{2} + 12 x + 4}{2 x + 1}$

Divide $4 x^{4} - 12 x^{3} + x^{2} + 12 x + 4$ 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 $0$ .

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

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

$2 x^{3}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

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

$2 x^{3}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

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

$2 x^{3}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

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

$2 x^{3}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

Divide the highest order term in the dividend $- 14 x^{3}$ by the highest order term in divisor $2 x$ .

$2 x^{3}$

$7 x^{2}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$7 x^{2}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x^{3} - 7 x^{2}$

$2 x^{3}$

$7 x^{2}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

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

$2 x^{3}$

$7 x^{2}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

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

$2 x^{3}$

$7 x^{2}$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

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

$2 x^{3}$

$7 x^{2}$

$4 x$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$7 x^{2}$

$4 x$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} + 4 x$

$2 x^{3}$

$7 x^{2}$

$4 x$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$4 x$

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

$2 x^{3}$

$7 x^{2}$

$4 x$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$4 x$

$8 x$

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

$2 x^{3}$

$7 x^{2}$

$4 x$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$4 x$

$8 x$

$4$

Divide the highest order term in the dividend $8 x$ by the highest order term in divisor $2 x$ .

$2 x^{3}$

$7 x^{2}$

$4 x$

$4$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$4 x$

$8 x$

$4$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$7 x^{2}$

$4 x$

$4$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$4 x$

$8 x$

$4$

$8 x$

$4$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x + 4$

$2 x^{3}$

$7 x^{2}$

$4 x$

$4$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$4 x$

$8 x$

$4$

$8 x$

$4$

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

$2 x^{3}$

$7 x^{2}$

$4 x$

$4$

$2 x$

$1$

$4 x^{4}$

$12 x^{3}$

$x^{2}$

$12 x$

$4$

$4 x^{4}$

$2 x^{3}$

$14 x^{3}$

$x^{2}$

$14 x^{3}$

$7 x^{2}$

$8 x^{2}$

$12 x$

$8 x^{2}$

$4 x$

$8 x$

$4$

$8 x$

$4$

$0$

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

$2 x^{3} - 7 x^{2} + 4 x + 4$

Write $4 x^{4} - 12 x^{3} + x^{2} + 12 x + 4$ as a set of factors.

$(2 x + 1) (2 x^{3} - 7 x^{2} + 4 x + 4) = 0$

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

$p = \pm 1, \pm 4, \pm 2 q = \pm 1, \pm 2$

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

$\pm 1, \pm 0.5, \pm 4, \pm 2$

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.

$2 {(- 0.5)}^{3} - 7 {(- 0.5)}^{2} + 4 \cdot - 0.5 + 4$

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

$2 \cdot - 0.125 - 7 {(- 0.5)}^{2} + 4 \cdot - 0.5 + 4$

Multiply $2$ by $- 0.125$ .

$- 0.25 - 7 {(- 0.5)}^{2} + 4 \cdot - 0.5 + 4$

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

$- 0.25 - 7 \cdot 0.25 + 4 \cdot - 0.5 + 4$

Multiply $- 7$ by $0.25$ .

$- 0.25 - 1.75 + 4 \cdot - 0.5 + 4$

Subtract $1.75$ from $- 0.25$ .

$- 2 + 4 \cdot - 0.5 + 4$

Multiply $4$ by $- 0.5$ .

$- 2 - 2 + 4$

Subtract $2$ from $- 2$ .

$- 4 + 4$

Add $- 4$ and $4$ .

$0$

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.

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

Divide $2 x^{3} - 7 x^{2} + 4 x + 4$ 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 $0$ .

$2 x$

$1$

$2 x^{3}$

$7 x^{2}$

$4 x$

$4$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$

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

				$x^{2}$	-	$4 x$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$
					-	$8 x^{2}$	-	$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} - 4 x$

				$x^{2}$	-	$4 x$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$
					+	$8 x^{2}$	+	$4 x$

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

				$x^{2}$	-	$4 x$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$
					+	$8 x^{2}$	+	$4 x$
							+	$8 x$

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

				$x^{2}$	-	$4 x$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$
					+	$8 x^{2}$	+	$4 x$
							+	$8 x$	+	$4$

Divide the highest order term in the dividend $8 x$ by the highest order term in divisor $2 x$ .

				$x^{2}$	-	$4 x$	+	$4$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$
					+	$8 x^{2}$	+	$4 x$
							+	$8 x$	+	$4$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$4$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$
					+	$8 x^{2}$	+	$4 x$
							+	$8 x$	+	$4$
							+	$8 x$	+	$4$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x + 4$

				$x^{2}$	-	$4 x$	+	$4$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$
					+	$8 x^{2}$	+	$4 x$
							+	$8 x$	+	$4$
							-	$8 x$	-	$4$

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

				$x^{2}$	-	$4 x$	+	$4$
$2 x$	+	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$4 x$	+	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$8 x^{2}$	+	$4 x$
					+	$8 x^{2}$	+	$4 x$
							+	$8 x$	+	$4$
							-	$8 x$	-	$4$
										$0$

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

$x^{2} - 4 x + 4$

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

$(2 x + 1) ((2 x + 1) (x^{2} - 4 x + 4)) = 0$

Factor using the perfect square rule.

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

$(2 x + 1) ((2 x + 1) (x^{2} - 4 x + 2^{2})) = 0$

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

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

Rewrite the polynomial.

$(2 x + 1) ((2 x + 1) (x^{2} - 2 \cdot x \cdot 2 + 2^{2})) = 0$

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

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

Remove unnecessary parentheses.

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

Combine like factors.

Raise $2 x + 1$ to the power of $1$ .

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

Raise $2 x + 1$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

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

${(2 x + 1)}^{2} = 0$

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

Set ${(2 x + 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(2 x + 1)}^{2}$ equal to $0$ .

${(2 x + 1)}^{2} = 0$

Solve ${(2 x + 1)}^{2} = 0$ for $x$ .

Set the $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve for $x$ .

Subtract $1$ from 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 $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

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

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

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

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

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

Step 3

Identify the Zeros and Their Multiplicities 4x^4-12x^3+x^2+12x+4

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