Find the Roots/Zeros Using the Rational Roots Test x^4-4x^3-5x^2+16x+4=0

$x^{4} - 4 x^{3} - 5 x^{2} + 16 x + 4 = 0$

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.

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

$q = \pm 1$

Step 2

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

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

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.

${(2)}^{4} - 4 {(2)}^{3} - 5 {(2)}^{2} + 16 (2) + 4$

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 $4$ .

$16 - 4 {(2)}^{3} - 5 {(2)}^{2} + 16 (2) + 4$

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

$16 - 4 \cdot 8 - 5 {(2)}^{2} + 16 (2) + 4$

Multiply $- 4$ by $8$ .

$16 - 32 - 5 {(2)}^{2} + 16 (2) + 4$

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

$16 - 32 - 5 \cdot 4 + 16 (2) + 4$

Multiply $- 5$ by $4$ .

$16 - 32 - 20 + 16 (2) + 4$

Multiply $16$ by $2$ .

$16 - 32 - 20 + 32 + 4$

Simplify by adding and subtracting.

Subtract $32$ from $16$ .

$- 16 - 20 + 32 + 4$

Subtract $20$ from $- 16$ .

$- 36 + 32 + 4$

Add $- 36$ and $32$ .

$- 4 + 4$

Add $- 4$ and $4$ .

$0$

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.

$\frac{x^{4} - 4 x^{3} - 5 x^{2} + 16 x + 4}{x - 2}$

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.

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\xa0				\xa0

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

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\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 $(- 4)$ .

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\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.

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\xa0	$2$			\xa0
	$1$	$- 2$

Multiply the newest entry in the result $(- 2)$ by the divisor $(2)$ and place the result of $(- 4)$ under the next term in the dividend $(- 5)$ .

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\xa0	$2$	$- 4$		\xa0
	$1$	$- 2$

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

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\xa0	$2$	$- 4$		\xa0
	$1$	$- 2$	$- 9$

Multiply the newest entry in the result $(- 9)$ by the divisor $(2)$ and place the result of $(- 18)$ under the next term in the dividend $(16)$ .

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\xa0	$2$	$- 4$	$- 18$	\xa0
	$1$	$- 2$	$- 9$

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

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\xa0	$2$	$- 4$	$- 18$	\xa0
	$1$	$- 2$	$- 9$	$- 2$

Multiply the newest entry in the result $(- 2)$ by the divisor $(2)$ and place the result of $(- 4)$ under the next term in the dividend $(4)$ .

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\xa0	$2$	$- 4$	$- 18$	$- 4$
	$1$	$- 2$	$- 9$	$- 2$

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

$2$	$1$	$- 4$	$- 5$	$16$	$4$
	\xa0	$2$	$- 4$	$- 18$	$- 4$
	$1$	$- 2$	$- 9$	$- 2$	$0$

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

$1 x^{3} + - 2 x^{2} + (- 9) x - 2$

Simplify the quotient polynomial.

$x^{3} - 2 x^{2} - 9 x - 2$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 2 x^{2} - 9 x - 2$ 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.

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

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

$\pm 1, \pm 2$

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - 2 {(- 2)}^{2} - 9 \cdot - 2 - 2$

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

$- 8 - 2 {(- 2)}^{2} - 9 \cdot - 2 - 2$

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

$- 8 - 2 \cdot 4 - 9 \cdot - 2 - 2$

Multiply $- 2$ by $4$ .

$- 8 - 8 - 9 \cdot - 2 - 2$

Subtract $8$ from $- 8$ .

$- 16 - 9 \cdot - 2 - 2$

Multiply $- 9$ by $- 2$ .

$- 16 + 18 - 2$

Add $- 16$ and $18$ .

$2 - 2$

Subtract $2$ from $2$ .

$0$

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.

$\frac{x^{3} - 2 x^{2} - 9 x - 2}{x + 2}$

Divide $x^{3} - 2 x^{2} - 9 x - 2$ by $x + 2$ .

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

$x$

$2$

$x^{3}$

$2 x^{2}$

$9 x$

$2$

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			+	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$
					-	$4 x^{2}$	-	$8 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$
					+	$4 x^{2}$	+	$8 x$

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

				$x^{2}$	-	$4 x$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$
					+	$4 x^{2}$	+	$8 x$
							-	$x$

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

				$x^{2}$	-	$4 x$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$
					+	$4 x^{2}$	+	$8 x$
							-	$x$	-	$2$

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

				$x^{2}$	-	$4 x$	-	$1$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$
					+	$4 x^{2}$	+	$8 x$
							-	$x$	-	$2$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	-	$1$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$
					+	$4 x^{2}$	+	$8 x$
							-	$x$	-	$2$
							-	$x$	-	$2$

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

				$x^{2}$	-	$4 x$	-	$1$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$
					+	$4 x^{2}$	+	$8 x$
							-	$x$	-	$2$
							+	$x$	+	$2$

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

				$x^{2}$	-	$4 x$	-	$1$
$x$	+	$2$		$x^{3}$	-	$2 x^{2}$	-	$9 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	-	$9 x$
					+	$4 x^{2}$	+	$8 x$
							-	$x$	-	$2$
							+	$x$	+	$2$
										$0$

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

$x^{2} - 4 x - 1$

Write $x^{3} - 2 x^{2} - 9 x - 2$ as a set of factors.

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

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

$x + 2 = 0$

$x^{2} - 4 x - 1 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x^{2} - 4 x - 1$ equal to $0$ and solve for $x$ .

Set $x^{2} - 4 x - 1$ equal to $0$ .

$x^{2} - 4 x - 1 = 0$

Solve $x^{2} - 4 x - 1 = 0$ 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 $c = - 1$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Add $16$ and $4$ .

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

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$x = \frac{4 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{5}}{2}$ .

$x = 2 \pm \sqrt{5}$

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Add $16$ and $4$ .

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

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$x = \frac{4 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{5}}{2}$ .

$x = 2 \pm \sqrt{5}$

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{5}$

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Add $16$ and $4$ .

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

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$x = \frac{4 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{5}}{2}$ .

$x = 2 \pm \sqrt{5}$

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{5}$

The final answer is the combination of both solutions.

$x = 2 + \sqrt{5}, 2 - \sqrt{5}$

The final solution is all the values that make $(x + 2) (x^{2} - 4 x - 1) = 0$ true.

$x = - 2, 2 + \sqrt{5}, 2 - \sqrt{5}$

Step 8

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

$(x - 2) (x + 2) (x - 2 - \sqrt{5}) (x - 2 + \sqrt{5})$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 4 x^{3} - 5 x^{2} + 16 x + 4$ .

$x = 2, - 2, 2 + \sqrt{5}, 2 - \sqrt{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 2, - 2, 2 + \sqrt{5}, 2 - \sqrt{5}$

Decimal Form:

$x = 2, - 2, 4.23606797 \dots, - 0.23606797 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test x^4-4x^3-5x^2+16x+4=0

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