Find the Roots/Zeros Using the Rational Roots Test x^4-4x^3-18x^2+108x-135

$x^{4} - 4 x^{3} - 18 x^{2} + 108 x - 135$

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.

$p = \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 27, \pm 45, \pm 135$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 27, \pm 45, \pm 135$

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.

${(3)}^{4} - 4 {(3)}^{3} - 18 {(3)}^{2} + 108 (3) - 135$

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 $4$ .

$81 - 4 {(3)}^{3} - 18 {(3)}^{2} + 108 (3) - 135$

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

$81 - 4 \cdot 27 - 18 {(3)}^{2} + 108 (3) - 135$

Multiply $- 4$ by $27$ .

$81 - 108 - 18 {(3)}^{2} + 108 (3) - 135$

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

$81 - 108 - 18 \cdot 9 + 108 (3) - 135$

Multiply $- 18$ by $9$ .

$81 - 108 - 162 + 108 (3) - 135$

Multiply $108$ by $3$ .

$81 - 108 - 162 + 324 - 135$

Simplify by adding and subtracting.

Subtract $108$ from $81$ .

$- 27 - 162 + 324 - 135$

Subtract $162$ from $- 27$ .

$- 189 + 324 - 135$

Add $- 189$ and $324$ .

$135 - 135$

Subtract $135$ from $135$ .

$0$

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.

$\frac{x^{4} - 4 x^{3} - 18 x^{2} + 108 x - 135}{x - 3}$

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.

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0				\xa0

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

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(3)$ and place the result of $(3)$ under the next term in the dividend $(- 4)$ .

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0	$3$			\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.

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0	$3$			\xa0
	$1$	$- 1$

Multiply the newest entry in the result $(- 1)$ by the divisor $(3)$ and place the result of $(- 3)$ under the next term in the dividend $(- 18)$ .

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0	$3$	$- 3$		\xa0
	$1$	$- 1$

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

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0	$3$	$- 3$		\xa0
	$1$	$- 1$	$- 21$

Multiply the newest entry in the result $(- 21)$ by the divisor $(3)$ and place the result of $(- 63)$ under the next term in the dividend $(108)$ .

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0	$3$	$- 3$	$- 63$	\xa0
	$1$	$- 1$	$- 21$

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

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0	$3$	$- 3$	$- 63$	\xa0
	$1$	$- 1$	$- 21$	$45$

Multiply the newest entry in the result $(45)$ by the divisor $(3)$ and place the result of $(135)$ under the next term in the dividend $(- 135)$ .

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0	$3$	$- 3$	$- 63$	$135$
	$1$	$- 1$	$- 21$	$45$

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

$3$	$1$	$- 4$	$- 18$	$108$	$- 135$
	\xa0	$3$	$- 3$	$- 63$	$135$
	$1$	$- 1$	$- 21$	$45$	$0$

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

$1 x^{3} + - 1 x^{2} + (- 21) x + 45$

Simplify the quotient polynomial.

$x^{3} - x^{2} - 21 x + 45$

Step 7

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $x^{3} - x^{2} - 21 x + 45$ 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.

$p = \pm 1, \pm 45, \pm 3, \pm 15, \pm 5, \pm 9 q = \pm 1$

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

$\pm 1, \pm 45, \pm 3, \pm 15, \pm 5, \pm 9$

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.

$3^{3} - 3^{2} - 21 \cdot 3 + 45$

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

$27 - 3^{2} - 21 \cdot 3 + 45$

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

$27 - 1 \cdot 9 - 21 \cdot 3 + 45$

Multiply $- 1$ by $9$ .

$27 - 9 - 21 \cdot 3 + 45$

Subtract $9$ from $27$ .

$18 - 21 \cdot 3 + 45$

Multiply $- 21$ by $3$ .

$18 - 63 + 45$

Subtract $63$ from $18$ .

$- 45 + 45$

Add $- 45$ and $45$ .

$0$

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.

$\frac{x^{3} - x^{2} - 21 x + 45}{x - 3}$

Divide $x^{3} - x^{2} - 21 x + 45$ 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 $0$ .

$x$

$3$

$x^{3}$

$x^{2}$

$21 x$

$45$

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			+	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$
					+	$2 x^{2}$	-	$6 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$
					-	$2 x^{2}$	+	$6 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$
					-	$2 x^{2}$	+	$6 x$
							-	$15 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$
					-	$2 x^{2}$	+	$6 x$
							-	$15 x$	+	$45$

Divide the highest order term in the dividend $- 15 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$
					-	$2 x^{2}$	+	$6 x$
							-	$15 x$	+	$45$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$
					-	$2 x^{2}$	+	$6 x$
							-	$15 x$	+	$45$
							-	$15 x$	+	$45$

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x + 45$

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$
					-	$2 x^{2}$	+	$6 x$
							-	$15 x$	+	$45$
							+	$15 x$	-	$45$

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

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$21 x$	+	$45$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$21 x$
					-	$2 x^{2}$	+	$6 x$
							-	$15 x$	+	$45$
							+	$15 x$	-	$45$
										$0$

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

$x^{2} + 2 x - 15$

Write $x^{3} - x^{2} - 21 x + 45$ as a set of factors.

$(x - 3) (x^{2} + 2 x - 15) = 0$

Factor $x^{2} + 2 x - 15$ 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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Combine like factors.

Raise $x - 3$ to the power of $1$ .

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

Raise $x - 3$ to the power of $1$ .

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

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

${(x - 3)}^{1 + 1} (x + 5) = 0$

Add $1$ and $1$ .

${(x - 3)}^{2} (x + 5) = 0$

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

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

$x + 5 = 0$

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make ${(x - 3)}^{2} (x + 5) = 0$ true.

$x = 3, - 5$

Step 8

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

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

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 4 x^{3} - 18 x^{2} + 108 x - 135$ .

$x = 3, - 5$

Step 10

Find the Roots/Zeros Using the Rational Roots Test x^4-4x^3-18x^2+108x-135

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