Find the Roots/Zeros Using the Rational Roots Test 4x^4-21x^3-46x^2+219x+180

$4 x^{4} - 21 x^{3} - 46 x^{2} + 219 x + 180$

Step 1

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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 9, \pm 10, \pm 12, \pm 15, \pm 18, \pm 20, \pm 30, \pm 36, \pm 45, \pm 60, \pm 90, \pm 180$

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

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 4, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}, \pm 6, \pm 9, \pm \frac{9}{2}, \pm \frac{9}{4}, \pm 10, \pm 12, \pm 15, \pm \frac{15}{2}, \pm \frac{15}{4}, \pm 18, \pm 20, \pm 30, \pm 36, \pm 45, \pm \frac{45}{2}, \pm \frac{45}{4}, \pm 60, \pm 90, \pm 180$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$4 {(- 3)}^{4} - 21 {(- 3)}^{3} - 46 {(- 3)}^{2} + 219 (- 3) + 180$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 3$ is a root of the polynomial.

Simplify each term.

Raise $- 3$ to the power of $4$ .

$4 \cdot 81 - 21 {(- 3)}^{3} - 46 {(- 3)}^{2} + 219 (- 3) + 180$

Multiply $4$ by $81$ .

$324 - 21 {(- 3)}^{3} - 46 {(- 3)}^{2} + 219 (- 3) + 180$

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

$324 - 21 \cdot - 27 - 46 {(- 3)}^{2} + 219 (- 3) + 180$

Multiply $- 21$ by $- 27$ .

$324 + 567 - 46 {(- 3)}^{2} + 219 (- 3) + 180$

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

$324 + 567 - 46 \cdot 9 + 219 (- 3) + 180$

Multiply $- 46$ by $9$ .

$324 + 567 - 414 + 219 (- 3) + 180$

Multiply $219$ by $- 3$ .

$324 + 567 - 414 - 657 + 180$

Simplify by adding and subtracting.

Add $324$ and $567$ .

$891 - 414 - 657 + 180$

Subtract $414$ from $891$ .

$477 - 657 + 180$

Subtract $657$ from $477$ .

$- 180 + 180$

Add $- 180$ and $180$ .

$0$

Step 5

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{4} - 21 x^{3} - 46 x^{2} + 219 x + 180}{x + 3}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0				\xa0

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0				\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- 3)$ and place the result of $(- 12)$ under the next term in the dividend $(- 21)$ .

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0	$- 12$			\xa0
	$4$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0	$- 12$			\xa0
	$4$	$- 33$

Multiply the newest entry in the result $(- 33)$ by the divisor $(- 3)$ and place the result of $(99)$ under the next term in the dividend $(- 46)$ .

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0	$- 12$	$99$		\xa0
	$4$	$- 33$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0	$- 12$	$99$		\xa0
	$4$	$- 33$	$53$

Multiply the newest entry in the result $(53)$ by the divisor $(- 3)$ and place the result of $(- 159)$ under the next term in the dividend $(219)$ .

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0	$- 12$	$99$	$- 159$	\xa0
	$4$	$- 33$	$53$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0	$- 12$	$99$	$- 159$	\xa0
	$4$	$- 33$	$53$	$60$

Multiply the newest entry in the result $(60)$ by the divisor $(- 3)$ and place the result of $(- 180)$ under the next term in the dividend $(180)$ .

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0	$- 12$	$99$	$- 159$	$- 180$
	$4$	$- 33$	$53$	$60$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$4$	$- 21$	$- 46$	$219$	$180$
	\xa0	$- 12$	$99$	$- 159$	$- 180$
	$4$	$- 33$	$53$	$60$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$4 x^{3} + - 33 x^{2} + (53) x + 60$

Simplify the quotient polynomial.

$4 x^{3} - 33 x^{2} + 53 x + 60$

Step 7

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $4 x^{3} - 33 x^{2} + 53 x + 60$ 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.

$p = \pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10 q = \pm 1, \pm 4, \pm 2$

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 60, \pm 15, \pm 30, \pm 2, \pm 7.5, \pm 3, \pm 0.75, \pm 1.5, \pm 20, \pm 5, \pm 10, \pm 4, \pm 3.75, \pm 1.25, \pm 2.5, \pm 12, \pm 6$

Substitute $- 0.75$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.75$ is a root of the polynomial.

Substitute $- 0.75$ into the polynomial.

$4 {(- 0.75)}^{3} - 33 {(- 0.75)}^{2} + 53 \cdot - 0.75 + 60$

Raise $- 0.75$ to the power of $3$ .

$4 \cdot - 0.421875 - 33 {(- 0.75)}^{2} + 53 \cdot - 0.75 + 60$

Multiply $4$ by $- 0.421875$ .

$- 1.6875 - 33 {(- 0.75)}^{2} + 53 \cdot - 0.75 + 60$

Raise $- 0.75$ to the power of $2$ .

$- 1.6875 - 33 \cdot 0.5625 + 53 \cdot - 0.75 + 60$

Multiply $- 33$ by $0.5625$ .

$- 1.6875 - 18.5625 + 53 \cdot - 0.75 + 60$

Subtract $18.5625$ from $- 1.6875$ .

$- 20.25 + 53 \cdot - 0.75 + 60$

Multiply $53$ by $- 0.75$ .

$- 20.25 - 39.75 + 60$

Subtract $39.75$ from $- 20.25$ .

$- 60 + 60$

Add $- 60$ and $60$ .

$0$

Since $- 0.75$ is a known root, divide the polynomial by $4 x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 33 x^{2} + 53 x + 60}{4 x + 3}$

Divide $4 x^{3} - 33 x^{2} + 53 x + 60$ by $4 x + 3$ .

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

$4 x$

$3$

$4 x^{3}$

$33 x^{2}$

$53 x$

$60$

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $4 x$ .

					$x^{2}$
	$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$

Multiply the new quotient term by the divisor.

				$x^{2}$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			+	$4 x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$

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

				$x^{2}$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$

Divide the highest order term in the dividend $- 36 x^{2}$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$9 x$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$
					-	$36 x^{2}$	-	$27 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 36 x^{2} - 27 x$

				$x^{2}$	-	$9 x$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$
					+	$36 x^{2}$	+	$27 x$

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

				$x^{2}$	-	$9 x$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$
					+	$36 x^{2}$	+	$27 x$
							+	$80 x$

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

				$x^{2}$	-	$9 x$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$
					+	$36 x^{2}$	+	$27 x$
							+	$80 x$	+	$60$

Divide the highest order term in the dividend $80 x$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$9 x$	+	$20$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$
					+	$36 x^{2}$	+	$27 x$
							+	$80 x$	+	$60$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$9 x$	+	$20$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$
					+	$36 x^{2}$	+	$27 x$
							+	$80 x$	+	$60$
							+	$80 x$	+	$60$

The expression needs to be subtracted from the dividend, so change all the signs in $80 x + 60$

				$x^{2}$	-	$9 x$	+	$20$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$
					+	$36 x^{2}$	+	$27 x$
							+	$80 x$	+	$60$
							-	$80 x$	-	$60$

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

				$x^{2}$	-	$9 x$	+	$20$
$4 x$	+	$3$		$4 x^{3}$	-	$33 x^{2}$	+	$53 x$	+	$60$
			-	$4 x^{3}$	-	$3 x^{2}$
					-	$36 x^{2}$	+	$53 x$
					+	$36 x^{2}$	+	$27 x$
							+	$80 x$	+	$60$
							-	$80 x$	-	$60$
										$0$

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

$x^{2} - 9 x + 20$

Write $4 x^{3} - 33 x^{2} + 53 x + 60$ as a set of factors.

$(4 x + 3) (x^{2} - 9 x + 20) = 0$

Factor $x^{2} - 9 x + 20$ 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 $20$ and whose sum is $- 9$ .

$- 5, - 4$

Write the factored form using these integers.

$(4 x + 3) ((x - 5) (x - 4)) = 0$

Remove unnecessary parentheses.

$(4 x + 3) (x - 5) (x - 4) = 0$

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

$4 x + 3 = 0$

$x - 5 = 0$

$x - 4 = 0$

Set $4 x + 3$ equal to $0$ and solve for $x$ .

Set $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Solve $4 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Divide each term in $4 x = - 3$ by $4$ and simplify.

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(4 x + 3) (x - 5) (x - 4) = 0$ true.

$x = - \frac{3}{4}, 5, 4$

Step 8

The polynomial can be written as a set of linear factors.

$(x + 3) (x + \frac{3}{4}) (x - 5) (x - 4)$

Step 9

These are the roots (zeros) of the polynomial $4 x^{4} - 21 x^{3} - 46 x^{2} + 219 x + 180$ .

$x = - 3, - \frac{3}{4}, 5, 4$

Step 10

Find the Roots/Zeros Using the Rational Roots Test 4x^4-21x^3-46x^2+219x+180

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