Find the Roots/Zeros Using the Rational Roots Test x^4-6x^3-19x^2+84x+180

$x^{4} - 6 x^{3} - 19 x^{2} + 84 x + 180$

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.

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

$q = \pm 1$

Step 2

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

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

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.

${(- 2)}^{4} - 6 {(- 2)}^{3} - 19 {(- 2)}^{2} + 84 (- 2) + 180$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 2$ is a root of the polynomial.

Simplify each term.

Raise $- 2$ to the power of $4$ .

$16 - 6 {(- 2)}^{3} - 19 {(- 2)}^{2} + 84 (- 2) + 180$

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

$16 - 6 \cdot - 8 - 19 {(- 2)}^{2} + 84 (- 2) + 180$

Multiply $- 6$ by $- 8$ .

$16 + 48 - 19 {(- 2)}^{2} + 84 (- 2) + 180$

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

$16 + 48 - 19 \cdot 4 + 84 (- 2) + 180$

Multiply $- 19$ by $4$ .

$16 + 48 - 76 + 84 (- 2) + 180$

Multiply $84$ by $- 2$ .

$16 + 48 - 76 - 168 + 180$

Simplify by adding and subtracting.

Add $16$ and $48$ .

$64 - 76 - 168 + 180$

Subtract $76$ from $64$ .

$- 12 - 168 + 180$

Subtract $168$ from $- 12$ .

$- 180 + 180$

Add $- 180$ and $180$ .

$0$

Step 5

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.

$\frac{x^{4} - 6 x^{3} - 19 x^{2} + 84 x + 180}{x + 2}$

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.

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 2)$ and place the result of $(- 2)$ under the next term in the dividend $(- 6)$ .

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0	$- 2$			\xa0
	$1$

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

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0	$- 2$			\xa0
	$1$	$- 8$

Multiply the newest entry in the result $(- 8)$ by the divisor $(- 2)$ and place the result of $(16)$ under the next term in the dividend $(- 19)$ .

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0	$- 2$	$16$		\xa0
	$1$	$- 8$

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

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0	$- 2$	$16$		\xa0
	$1$	$- 8$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 2)$ and place the result of $(6)$ under the next term in the dividend $(84)$ .

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0	$- 2$	$16$	$6$	\xa0
	$1$	$- 8$	$- 3$

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

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0	$- 2$	$16$	$6$	\xa0
	$1$	$- 8$	$- 3$	$90$

Multiply the newest entry in the result $(90)$ by the divisor $(- 2)$ and place the result of $(- 180)$ under the next term in the dividend $(180)$ .

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0	$- 2$	$16$	$6$	$- 180$
	$1$	$- 8$	$- 3$	$90$

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

$- 2$	$1$	$- 6$	$- 19$	$84$	$180$
	\xa0	$- 2$	$16$	$6$	$- 180$
	$1$	$- 8$	$- 3$	$90$	$0$

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

$1 x^{3} + - 8 x^{2} + (- 3) x + 90$

Simplify the quotient polynomial.

$x^{3} - 8 x^{2} - 3 x + 90$

Step 7

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $x^{3} - 8 x^{2} - 3 x + 90$ 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.

$p = \pm 1, \pm 90, \pm 2, \pm 45, \pm 3, \pm 30, \pm 5, \pm 18, \pm 6, \pm 15, \pm 9, \pm 10 q = \pm 1$

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

$\pm 1, \pm 90, \pm 2, \pm 45, \pm 3, \pm 30, \pm 5, \pm 18, \pm 6, \pm 15, \pm 9, \pm 10$

Substitute $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

Substitute $- 3$ into the polynomial.

${(- 3)}^{3} - 8 {(- 3)}^{2} - 3 \cdot - 3 + 90$

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

$- 27 - 8 {(- 3)}^{2} - 3 \cdot - 3 + 90$

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

$- 27 - 8 \cdot 9 - 3 \cdot - 3 + 90$

Multiply $- 8$ by $9$ .

$- 27 - 72 - 3 \cdot - 3 + 90$

Subtract $72$ from $- 27$ .

$- 99 - 3 \cdot - 3 + 90$

Multiply $- 3$ by $- 3$ .

$- 99 + 9 + 90$

Add $- 99$ and $9$ .

$- 90 + 90$

Add $- 90$ and $90$ .

$0$

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.

$\frac{x^{3} - 8 x^{2} - 3 x + 90}{x + 3}$

Divide $x^{3} - 8 x^{2} - 3 x + 90$ by $x + 3$ .

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

$x$

$3$

$x^{3}$

$8 x^{2}$

$3 x$

$90$

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			+	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$

Divide the highest order term in the dividend $- 11 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$11 x$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$11 x$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$
					-	$11 x^{2}$	-	$33 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 11 x^{2} - 33 x$

				$x^{2}$	-	$11 x$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$
					+	$11 x^{2}$	+	$33 x$

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

				$x^{2}$	-	$11 x$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$
					+	$11 x^{2}$	+	$33 x$
							+	$30 x$

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

				$x^{2}$	-	$11 x$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$
					+	$11 x^{2}$	+	$33 x$
							+	$30 x$	+	$90$

Divide the highest order term in the dividend $30 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$11 x$	+	$30$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$
					+	$11 x^{2}$	+	$33 x$
							+	$30 x$	+	$90$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$11 x$	+	$30$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$
					+	$11 x^{2}$	+	$33 x$
							+	$30 x$	+	$90$
							+	$30 x$	+	$90$

The expression needs to be subtracted from the dividend, so change all the signs in $30 x + 90$

				$x^{2}$	-	$11 x$	+	$30$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$
					+	$11 x^{2}$	+	$33 x$
							+	$30 x$	+	$90$
							-	$30 x$	-	$90$

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

				$x^{2}$	-	$11 x$	+	$30$
$x$	+	$3$		$x^{3}$	-	$8 x^{2}$	-	$3 x$	+	$90$
			-	$x^{3}$	-	$3 x^{2}$
					-	$11 x^{2}$	-	$3 x$
					+	$11 x^{2}$	+	$33 x$
							+	$30 x$	+	$90$
							-	$30 x$	-	$90$
										$0$

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

$x^{2} - 11 x + 30$

Write $x^{3} - 8 x^{2} - 3 x + 90$ as a set of factors.

$(x + 3) (x^{2} - 11 x + 30) = 0$

Factor $x^{2} - 11 x + 30$ 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 $30$ and whose sum is $- 11$ .

$- 6, - 5$

Write the factored form using these integers.

$(x + 3) ((x - 6) (x - 5)) = 0$

Remove unnecessary parentheses.

$(x + 3) (x - 6) (x - 5) = 0$

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

$x + 3 = 0$

$x - 6 = 0$

$x - 5 = 0$

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.

$x = - 3$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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.

$x = 5$

The final solution is all the values that make $(x + 3) (x - 6) (x - 5) = 0$ true.

$x = - 3, 6, 5$

Step 8

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

$(x + 2) (x + 3) (x - 6) (x - 5)$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 6 x^{3} - 19 x^{2} + 84 x + 180$ .

$x = - 2, - 3, 6, 5$

Step 10

Find the Roots/Zeros Using the Rational Roots Test x^4-6x^3-19x^2+84x+180

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