Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-4x^3-4x^2+36x-45

$f (x) = x^{4} - 4 x^{3} - 4 x^{2} + 36 x - 45$

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.

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

$q = \pm 1$

Step 2

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

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

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.

${(3)}^{4} - 4 {(3)}^{3} - 4 {(3)}^{2} + 36 (3) - 45$

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 $4$ .

$81 - 4 {(3)}^{3} - 4 {(3)}^{2} + 36 (3) - 45$

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

$81 - 4 \cdot 27 - 4 {(3)}^{2} + 36 (3) - 45$

Multiply $- 4$ by $27$ .

$81 - 108 - 4 {(3)}^{2} + 36 (3) - 45$

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

$81 - 108 - 4 \cdot 9 + 36 (3) - 45$

Multiply $- 4$ by $9$ .

$81 - 108 - 36 + 36 (3) - 45$

Multiply $36$ by $3$ .

$81 - 108 - 36 + 108 - 45$

Simplify by adding and subtracting.

Subtract $108$ from $81$ .

$- 27 - 36 + 108 - 45$

Subtract $36$ from $- 27$ .

$- 63 + 108 - 45$

Add $- 63$ and $108$ .

$45 - 45$

Subtract $45$ from $45$ .

$0$

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.

$\frac{x^{4} - 4 x^{3} - 4 x^{2} + 36 x - 45}{x - 3}$

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.

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\xa0				\xa0

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

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\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 $(- 4)$ .

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\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.

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\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 $(- 4)$ .

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\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.

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\xa0	$3$	$- 3$		\xa0
	$1$	$- 1$	$- 7$

Multiply the newest entry in the result $(- 7)$ by the divisor $(3)$ and place the result of $(- 21)$ under the next term in the dividend $(36)$ .

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\xa0	$3$	$- 3$	$- 21$	\xa0
	$1$	$- 1$	$- 7$

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

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\xa0	$3$	$- 3$	$- 21$	\xa0
	$1$	$- 1$	$- 7$	$15$

Multiply the newest entry in the result $(15)$ by the divisor $(3)$ and place the result of $(45)$ under the next term in the dividend $(- 45)$ .

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\xa0	$3$	$- 3$	$- 21$	$45$
	$1$	$- 1$	$- 7$	$15$

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

$3$	$1$	$- 4$	$- 4$	$36$	$- 45$
	\xa0	$3$	$- 3$	$- 21$	$45$
	$1$	$- 1$	$- 7$	$15$	$0$

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

$1 x^{3} + - 1 x^{2} + (- 7) x + 15$

Simplify the quotient polynomial.

$x^{3} - x^{2} - 7 x + 15$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - x^{2} - 7 x + 15$ 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.

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

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

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

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.

${(- 3)}^{3} - {(- 3)}^{2} - 7 \cdot - 3 + 15$

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

$- 27 - {(- 3)}^{2} - 7 \cdot - 3 + 15$

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

$- 27 - 1 \cdot 9 - 7 \cdot - 3 + 15$

Multiply $- 1$ by $9$ .

$- 27 - 9 - 7 \cdot - 3 + 15$

Subtract $9$ from $- 27$ .

$- 36 - 7 \cdot - 3 + 15$

Multiply $- 7$ by $- 3$ .

$- 36 + 21 + 15$

Add $- 36$ and $21$ .

$- 15 + 15$

Add $- 15$ and $15$ .

$0$

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.

$\frac{x^{3} - x^{2} - 7 x + 15}{x + 3}$

Divide $x^{3} - x^{2} - 7 x + 15$ 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 $0$ .

$x$

$3$

$x^{3}$

$x^{2}$

$7 x$

$15$

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			+	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$
					-	$4 x^{2}$	-	$12 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} - 12 x$

				$x^{2}$	-	$4 x$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$
					+	$4 x^{2}$	+	$12 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$
					+	$4 x^{2}$	+	$12 x$
							+	$5 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$
					+	$4 x^{2}$	+	$12 x$
							+	$5 x$	+	$15$

Divide the highest order term in the dividend $5 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$5$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$
					+	$4 x^{2}$	+	$12 x$
							+	$5 x$	+	$15$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$5$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$
					+	$4 x^{2}$	+	$12 x$
							+	$5 x$	+	$15$
							+	$5 x$	+	$15$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x + 15$

				$x^{2}$	-	$4 x$	+	$5$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$
					+	$4 x^{2}$	+	$12 x$
							+	$5 x$	+	$15$
							-	$5 x$	-	$15$

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

				$x^{2}$	-	$4 x$	+	$5$
$x$	+	$3$		$x^{3}$	-	$x^{2}$	-	$7 x$	+	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$4 x^{2}$	-	$7 x$
					+	$4 x^{2}$	+	$12 x$
							+	$5 x$	+	$15$
							-	$5 x$	-	$15$
										$0$

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

$x^{2} - 4 x + 5$

Write $x^{3} - x^{2} - 7 x + 15$ as a set of factors.

$(x + 3) (x^{2} - 4 x + 5) = 0$

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

$x + 3 = 0$

$x^{2} - 4 x + 5 = 0$

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 $x^{2} - 4 x + 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 4 x + 5$ equal to $0$ .

$x^{2} - 4 x + 5 = 0$

Solve $x^{2} - 4 x + 5 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Subtract $20$ from $16$ .

$x = \frac{4 \pm \sqrt{- 4}}{2 \cdot 1}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 2}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Subtract $20$ from $16$ .

$x = \frac{4 \pm \sqrt{- 4}}{2 \cdot 1}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 2}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

Change the $\pm$ to $+$ .

$x = 2 + i$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Subtract $20$ from $16$ .

$x = \frac{4 \pm \sqrt{- 4}}{2 \cdot 1}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 2}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

Change the $\pm$ to $-$ .

$x = 2 - i$

The final answer is the combination of both solutions.

$x = 2 + i, 2 - i$

The final solution is all the values that make $(x + 3) (x^{2} - 4 x + 5) = 0$ true.

$x = - 3, 2 + i, 2 - i$

Step 8

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

$(x - 3) (x + 3) (x - 2 - i) (x - 2 + i)$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 4 x^{3} - 4 x^{2} + 36 x - 45$ .

$x = 3, - 3, 2 + i, 2 - i$

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-4x^3-4x^2+36x-45

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