Find the Roots/Zeros Using the Rational Roots Test x^4-14x^3+34x^2+114x+65

$x^{4} - 14 x^{3} + 34 x^{2} + 114 x + 65$

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.

$p = \pm 1, \pm 5, \pm 13, \pm 65$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 5, \pm 13, \pm 65$

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.

${(- 1)}^{4} - 14 {(- 1)}^{3} + 34 {(- 1)}^{2} + 114 (- 1) + 65$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ is a root of the polynomial.

Simplify each term.

Raise $- 1$ to the power of $4$ .

$1 - 14 {(- 1)}^{3} + 34 {(- 1)}^{2} + 114 (- 1) + 65$

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

$1 - 14 \cdot - 1 + 34 {(- 1)}^{2} + 114 (- 1) + 65$

Multiply $- 14$ by $- 1$ .

$1 + 14 + 34 {(- 1)}^{2} + 114 (- 1) + 65$

Raise $- 1$ to the power of $2$ .

$1 + 14 + 34 \cdot 1 + 114 (- 1) + 65$

Multiply $34$ by $1$ .

$1 + 14 + 34 + 114 (- 1) + 65$

Multiply $114$ by $- 1$ .

$1 + 14 + 34 - 114 + 65$

Simplify by adding and subtracting.

Add $1$ and $14$ .

$15 + 34 - 114 + 65$

Add $15$ and $34$ .

$49 - 114 + 65$

Subtract $114$ from $49$ .

$- 65 + 65$

Add $- 65$ and $65$ .

$0$

Step 5

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 14 x^{3} + 34 x^{2} + 114 x + 65}{x + 1}$

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.

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0				\xa0

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

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 1)$ and place the result of $(- 1)$ under the next term in the dividend $(- 14)$ .

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0	$- 1$			\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.

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0	$- 1$			\xa0
	$1$	$- 15$

Multiply the newest entry in the result $(- 15)$ by the divisor $(- 1)$ and place the result of $(15)$ under the next term in the dividend $(34)$ .

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0	$- 1$	$15$		\xa0
	$1$	$- 15$

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

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0	$- 1$	$15$		\xa0
	$1$	$- 15$	$49$

Multiply the newest entry in the result $(49)$ by the divisor $(- 1)$ and place the result of $(- 49)$ under the next term in the dividend $(114)$ .

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0	$- 1$	$15$	$- 49$	\xa0
	$1$	$- 15$	$49$

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

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0	$- 1$	$15$	$- 49$	\xa0
	$1$	$- 15$	$49$	$65$

Multiply the newest entry in the result $(65)$ by the divisor $(- 1)$ and place the result of $(- 65)$ under the next term in the dividend $(65)$ .

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0	$- 1$	$15$	$- 49$	$- 65$
	$1$	$- 15$	$49$	$65$

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

$- 1$	$1$	$- 14$	$34$	$114$	$65$
	\xa0	$- 1$	$15$	$- 49$	$- 65$
	$1$	$- 15$	$49$	$65$	$0$

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

$1 x^{3} + - 15 x^{2} + (49) x + 65$

Simplify the quotient polynomial.

$x^{3} - 15 x^{2} + 49 x + 65$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 15 x^{2} + 49 x + 65$ 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.

$p = \pm 1, \pm 65, \pm 5, \pm 13 q = \pm 1$

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

$\pm 1, \pm 65, \pm 5, \pm 13$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 15 {(- 1)}^{2} + 49 \cdot - 1 + 65$

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

$- 1 - 15 {(- 1)}^{2} + 49 \cdot - 1 + 65$

Raise $- 1$ to the power of $2$ .

$- 1 - 15 \cdot 1 + 49 \cdot - 1 + 65$

Multiply $- 15$ by $1$ .

$- 1 - 15 + 49 \cdot - 1 + 65$

Subtract $15$ from $- 1$ .

$- 16 + 49 \cdot - 1 + 65$

Multiply $49$ by $- 1$ .

$- 16 - 49 + 65$

Subtract $49$ from $- 16$ .

$- 65 + 65$

Add $- 65$ and $65$ .

$0$

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 15 x^{2} + 49 x + 65}{x + 1}$

Divide $x^{3} - 15 x^{2} + 49 x + 65$ by $x + 1$ .

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

$x$

$1$

$x^{3}$

$15 x^{2}$

$49 x$

$65$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$

Divide the highest order term in the dividend $- 16 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$16 x$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$16 x$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$
					-	$16 x^{2}$	-	$16 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x^{2} - 16 x$

				$x^{2}$	-	$16 x$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$
					+	$16 x^{2}$	+	$16 x$

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

				$x^{2}$	-	$16 x$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$
					+	$16 x^{2}$	+	$16 x$
							+	$65 x$

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

				$x^{2}$	-	$16 x$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$
					+	$16 x^{2}$	+	$16 x$
							+	$65 x$	+	$65$

Divide the highest order term in the dividend $65 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$16 x$	+	$65$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$
					+	$16 x^{2}$	+	$16 x$
							+	$65 x$	+	$65$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$16 x$	+	$65$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$
					+	$16 x^{2}$	+	$16 x$
							+	$65 x$	+	$65$
							+	$65 x$	+	$65$

The expression needs to be subtracted from the dividend, so change all the signs in $65 x + 65$

				$x^{2}$	-	$16 x$	+	$65$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$
					+	$16 x^{2}$	+	$16 x$
							+	$65 x$	+	$65$
							-	$65 x$	-	$65$

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

				$x^{2}$	-	$16 x$	+	$65$
$x$	+	$1$		$x^{3}$	-	$15 x^{2}$	+	$49 x$	+	$65$
			-	$x^{3}$	-	$x^{2}$
					-	$16 x^{2}$	+	$49 x$
					+	$16 x^{2}$	+	$16 x$
							+	$65 x$	+	$65$
							-	$65 x$	-	$65$
										$0$

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

$x^{2} - 16 x + 65$

Write $x^{3} - 15 x^{2} + 49 x + 65$ as a set of factors.

$(x + 1) (x^{2} - 16 x + 65) = 0$

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

$x + 1 = 0$

$x^{2} - 16 x + 65 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x^{2} - 16 x + 65$ equal to $0$ and solve for $x$ .

Set $x^{2} - 16 x + 65$ equal to $0$ .

$x^{2} - 16 x + 65 = 0$

Solve $x^{2} - 16 x + 65 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 16$ , and $c = 65$ into the quadratic formula and solve for $x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot 65)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 65}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 65$ .

Multiply $- 4$ by $1$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 65}}{2 \cdot 1}$

Multiply $- 4$ by $65$ .

$x = \frac{16 \pm \sqrt{256 - 260}}{2 \cdot 1}$

Subtract $260$ from $256$ .

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

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

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

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

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

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

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

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

$x = \frac{16 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

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

$x = \frac{16 \pm i \cdot 2}{2 \cdot 1}$

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

$x = \frac{16 \pm 2 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{16 \pm 2 i}{2}$

Simplify $\frac{16 \pm 2 i}{2}$ .

$x = 8 \pm i$

The final answer is the combination of both solutions.

$x = 8 + i, 8 - i$

The final solution is all the values that make $(x + 1) (x^{2} - 16 x + 65) = 0$ true.

$x = - 1, 8 + i, 8 - i$

Step 8

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

$(x + 1) (x - 8 - i) (x - 8 + i)$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 14 x^{3} + 34 x^{2} + 114 x + 65$ .

$x = - 1, 8 + i, 8 - i$

Step 10

Find the Roots/Zeros Using the Rational Roots Test x^4-14x^3+34x^2+114x+65

Related Questions

Support LCMGCF.com by disabling your adblocker.