Find the Roots/Zeros Using the Rational Roots Test x^4-2x^3-34x^2-67x-30

$x^{4} - 2 x^{3} - 34 x^{2} - 67 x - 30$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30$

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.

${(- 2)}^{4} - 2 {(- 2)}^{3} - 34 {(- 2)}^{2} - 67 \cdot - 2 - 30$

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 $4$ .

$16 - 2 {(- 2)}^{3} - 34 {(- 2)}^{2} - 67 \cdot - 2 - 30$

Multiply $- 2$ by ${(- 2)}^{3}$ by adding the exponents.

Multiply $- 2$ by ${(- 2)}^{3}$ .

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

$16 + {(- 2)}^{1} {(- 2)}^{3} - 34 {(- 2)}^{2} - 67 \cdot - 2 - 30$

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

$16 + {(- 2)}^{1 + 3} - 34 {(- 2)}^{2} - 67 \cdot - 2 - 30$

Add $1$ and $3$ .

$16 + {(- 2)}^{4} - 34 {(- 2)}^{2} - 67 \cdot - 2 - 30$

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

$16 + 16 - 34 {(- 2)}^{2} - 67 \cdot - 2 - 30$

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

$16 + 16 - 34 \cdot 4 - 67 \cdot - 2 - 30$

Multiply $- 34$ by $4$ .

$16 + 16 - 136 - 67 \cdot - 2 - 30$

Multiply $- 67$ by $- 2$ .

$16 + 16 - 136 + 134 - 30$

Simplify by adding and subtracting.

Add $16$ and $16$ .

$32 - 136 + 134 - 30$

Subtract $136$ from $32$ .

$- 104 + 134 - 30$

Add $- 104$ and $134$ .

$30 - 30$

Subtract $30$ from $30$ .

$0$

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.

$\frac{x^{4} - 2 x^{3} - 34 x^{2} - 67 x - 30}{x + 2}$

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.

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\xa0				\xa0

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

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\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 $(- 2)$ .

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\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.

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\xa0	$- 2$			\xa0
	$1$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(- 2)$ and place the result of $(8)$ under the next term in the dividend $(- 34)$ .

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\xa0	$- 2$	$8$		\xa0
	$1$	$- 4$

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

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\xa0	$- 2$	$8$		\xa0
	$1$	$- 4$	$- 26$

Multiply the newest entry in the result $(- 26)$ by the divisor $(- 2)$ and place the result of $(52)$ under the next term in the dividend $(- 67)$ .

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\xa0	$- 2$	$8$	$52$	\xa0
	$1$	$- 4$	$- 26$

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

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\xa0	$- 2$	$8$	$52$	\xa0
	$1$	$- 4$	$- 26$	$- 15$

Multiply the newest entry in the result $(- 15)$ by the divisor $(- 2)$ and place the result of $(30)$ under the next term in the dividend $(- 30)$ .

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\xa0	$- 2$	$8$	$52$	$30$
	$1$	$- 4$	$- 26$	$- 15$

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

$- 2$	$1$	$- 2$	$- 34$	$- 67$	$- 30$
	\xa0	$- 2$	$8$	$52$	$30$
	$1$	$- 4$	$- 26$	$- 15$	$0$

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

$1 x^{3} + - 4 x^{2} + (- 26) x - 15$

Simplify the quotient polynomial.

$x^{3} - 4 x^{2} - 26 x - 15$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 4 x^{2} - 26 x - 15$ 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.

${(- 3)}^{3} - 4 {(- 3)}^{2} - 26 \cdot - 3 - 15$

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

$- 27 - 4 {(- 3)}^{2} - 26 \cdot - 3 - 15$

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

$- 27 - 4 \cdot 9 - 26 \cdot - 3 - 15$

Multiply $- 4$ by $9$ .

$- 27 - 36 - 26 \cdot - 3 - 15$

Subtract $36$ from $- 27$ .

$- 63 - 26 \cdot - 3 - 15$

Multiply $- 26$ by $- 3$ .

$- 63 + 78 - 15$

Add $- 63$ and $78$ .

$15 - 15$

Subtract $15$ from $15$ .

$0$

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.

$\frac{x^{3} - 4 x^{2} - 26 x - 15}{x + 3}$

Divide $x^{3} - 4 x^{2} - 26 x - 15$ 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 $0$ .

$x$

$3$

$x^{3}$

$4 x^{2}$

$26 x$

$15$

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			+	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$7 x$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$
					-	$7 x^{2}$	-	$21 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} - 21 x$

				$x^{2}$	-	$7 x$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$
					+	$7 x^{2}$	+	$21 x$

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

				$x^{2}$	-	$7 x$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$
					+	$7 x^{2}$	+	$21 x$
							-	$5 x$

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

				$x^{2}$	-	$7 x$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$
					+	$7 x^{2}$	+	$21 x$
							-	$5 x$	-	$15$

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

				$x^{2}$	-	$7 x$	-	$5$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$
					+	$7 x^{2}$	+	$21 x$
							-	$5 x$	-	$15$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$	-	$5$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$
					+	$7 x^{2}$	+	$21 x$
							-	$5 x$	-	$15$
							-	$5 x$	-	$15$

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

				$x^{2}$	-	$7 x$	-	$5$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$
					+	$7 x^{2}$	+	$21 x$
							-	$5 x$	-	$15$
							+	$5 x$	+	$15$

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

				$x^{2}$	-	$7 x$	-	$5$
$x$	+	$3$		$x^{3}$	-	$4 x^{2}$	-	$26 x$	-	$15$
			-	$x^{3}$	-	$3 x^{2}$
					-	$7 x^{2}$	-	$26 x$
					+	$7 x^{2}$	+	$21 x$
							-	$5 x$	-	$15$
							+	$5 x$	+	$15$
										$0$

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

$x^{2} - 7 x - 5$

Write $x^{3} - 4 x^{2} - 26 x - 15$ as a set of factors.

$(x + 3) (x^{2} - 7 x - 5) = 0$

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

$x + 3 = 0$

$x^{2} - 7 x - 5 = 0$

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 $x^{2} - 7 x - 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 7 x - 5$ equal to $0$ .

$x^{2} - 7 x - 5 = 0$

Solve $x^{2} - 7 x - 5 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 7$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4$ by $- 5$ .

$x = \frac{7 \pm \sqrt{49 + 20}}{2 \cdot 1}$

Add $49$ and $20$ .

$x = \frac{7 \pm \sqrt{69}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{7 \pm \sqrt{69}}{2}$

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{69}}{2}, \frac{7 - \sqrt{69}}{2}$

The final solution is all the values that make $(x + 3) (x^{2} - 7 x - 5) = 0$ true.

$x = - 3, \frac{7 + \sqrt{69}}{2}, \frac{7 - \sqrt{69}}{2}$

Step 8

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

$(x + 2) (x + 3) (x - \frac{7 + \sqrt{69}}{2}) (x - \frac{7 - \sqrt{69}}{2})$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 2 x^{3} - 34 x^{2} - 67 x - 30$ .

$x = - 2, - 3, \frac{7 + \sqrt{69}}{2}, \frac{7 - \sqrt{69}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 2, - 3, \frac{7 + \sqrt{69}}{2}, \frac{7 - \sqrt{69}}{2}$

Decimal Form:

$x = - 2, - 3, 7.65331193 \dots, - 0.65331193 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test x^4-2x^3-34x^2-67x-30

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