Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-6x^3-44x^2-58x-21

$f (x) = x^{4} - 6 x^{3} - 44 x^{2} - 58 x - 21$

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.

$p = \pm 1, \pm 3, \pm 7, \pm 21$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 3, \pm 7, \pm 21$

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.

${(- 1)}^{4} - 6 {(- 1)}^{3} - 44 {(- 1)}^{2} - 58 \cdot - 1 - 21$

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 $4$ .

$1 - 6 {(- 1)}^{3} - 44 {(- 1)}^{2} - 58 \cdot - 1 - 21$

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

$1 - 6 \cdot - 1 - 44 {(- 1)}^{2} - 58 \cdot - 1 - 21$

Multiply $- 6$ by $- 1$ .

$1 + 6 - 44 {(- 1)}^{2} - 58 \cdot - 1 - 21$

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

$1 + 6 - 44 \cdot 1 - 58 \cdot - 1 - 21$

Multiply $- 44$ by $1$ .

$1 + 6 - 44 - 58 \cdot - 1 - 21$

Multiply $- 58$ by $- 1$ .

$1 + 6 - 44 + 58 - 21$

Simplify by adding and subtracting.

Add $1$ and $6$ .

$7 - 44 + 58 - 21$

Subtract $44$ from $7$ .

$- 37 + 58 - 21$

Add $- 37$ and $58$ .

$21 - 21$

Subtract $21$ from $21$ .

$0$

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.

$\frac{x^{4} - 6 x^{3} - 44 x^{2} - 58 x - 21}{x + 1}$

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.

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\xa0				\xa0

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

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\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 $(- 6)$ .

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\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.

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\xa0	$- 1$			\xa0
	$1$	$- 7$

Multiply the newest entry in the result $(- 7)$ by the divisor $(- 1)$ and place the result of $(7)$ under the next term in the dividend $(- 44)$ .

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\xa0	$- 1$	$7$		\xa0
	$1$	$- 7$

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

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\xa0	$- 1$	$7$		\xa0
	$1$	$- 7$	$- 37$

Multiply the newest entry in the result $(- 37)$ by the divisor $(- 1)$ and place the result of $(37)$ under the next term in the dividend $(- 58)$ .

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\xa0	$- 1$	$7$	$37$	\xa0
	$1$	$- 7$	$- 37$

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

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\xa0	$- 1$	$7$	$37$	\xa0
	$1$	$- 7$	$- 37$	$- 21$

Multiply the newest entry in the result $(- 21)$ by the divisor $(- 1)$ and place the result of $(21)$ under the next term in the dividend $(- 21)$ .

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\xa0	$- 1$	$7$	$37$	$21$
	$1$	$- 7$	$- 37$	$- 21$

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

$- 1$	$1$	$- 6$	$- 44$	$- 58$	$- 21$
	\xa0	$- 1$	$7$	$37$	$21$
	$1$	$- 7$	$- 37$	$- 21$	$0$

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

$1 x^{3} + - 7 x^{2} + (- 37) x - 21$

Simplify the quotient polynomial.

$x^{3} - 7 x^{2} - 37 x - 21$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 7 x^{2} - 37 x - 21$ 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.

$p = \pm 1, \pm 21, \pm 3, \pm 7 q = \pm 1$

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

$\pm 1, \pm 21, \pm 3, \pm 7$

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.

${(- 3)}^{3} - 7 {(- 3)}^{2} - 37 \cdot - 3 - 21$

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

$- 27 - 7 {(- 3)}^{2} - 37 \cdot - 3 - 21$

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

$- 27 - 7 \cdot 9 - 37 \cdot - 3 - 21$

Multiply $- 7$ by $9$ .

$- 27 - 63 - 37 \cdot - 3 - 21$

Subtract $63$ from $- 27$ .

$- 90 - 37 \cdot - 3 - 21$

Multiply $- 37$ by $- 3$ .

$- 90 + 111 - 21$

Add $- 90$ and $111$ .

$21 - 21$

Subtract $21$ from $21$ .

$0$

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.

$\frac{x^{3} - 7 x^{2} - 37 x - 21}{x + 3}$

Divide $x^{3} - 7 x^{2} - 37 x - 21$ 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 $0$ .

$x$

$3$

$x^{3}$

$7 x^{2}$

$37 x$

$21$

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

					$x^{2}$
	$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			+	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$

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

				$x^{2}$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$
					-	$10 x^{2}$	-	$30 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 30 x$

				$x^{2}$	-	$10 x$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$
					+	$10 x^{2}$	+	$30 x$

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

				$x^{2}$	-	$10 x$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$
					+	$10 x^{2}$	+	$30 x$
							-	$7 x$

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

				$x^{2}$	-	$10 x$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$
					+	$10 x^{2}$	+	$30 x$
							-	$7 x$	-	$21$

Divide the highest order term in the dividend $- 7 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$	-	$7$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$
					+	$10 x^{2}$	+	$30 x$
							-	$7 x$	-	$21$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$	-	$7$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$
					+	$10 x^{2}$	+	$30 x$
							-	$7 x$	-	$21$
							-	$7 x$	-	$21$

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

				$x^{2}$	-	$10 x$	-	$7$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$
					+	$10 x^{2}$	+	$30 x$
							-	$7 x$	-	$21$
							+	$7 x$	+	$21$

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

				$x^{2}$	-	$10 x$	-	$7$
$x$	+	$3$		$x^{3}$	-	$7 x^{2}$	-	$37 x$	-	$21$
			-	$x^{3}$	-	$3 x^{2}$
					-	$10 x^{2}$	-	$37 x$
					+	$10 x^{2}$	+	$30 x$
							-	$7 x$	-	$21$
							+	$7 x$	+	$21$
										$0$

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

$x^{2} - 10 x - 7$

Write $x^{3} - 7 x^{2} - 37 x - 21$ as a set of factors.

$(x + 3) (x^{2} - 10 x - 7) = 0$

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

$x + 3 = 0$

$x^{2} - 10 x - 7 = 0$

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 $x^{2} - 10 x - 7$ equal to $0$ and solve for $x$ .

Set $x^{2} - 10 x - 7$ equal to $0$ .

$x^{2} - 10 x - 7 = 0$

Solve $x^{2} - 10 x - 7 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 10$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot - 7)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 7$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 7}}{2 \cdot 1}$

Multiply $- 4$ by $- 7$ .

$x = \frac{10 \pm \sqrt{100 + 28}}{2 \cdot 1}$

Add $100$ and $28$ .

$x = \frac{10 \pm \sqrt{128}}{2 \cdot 1}$

Rewrite $128$ as $8^{2} \cdot 2$ .

Factor $64$ out of $128$ .

$x = \frac{10 \pm \sqrt{64 (2)}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{10 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{10 \pm 8 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 8 \sqrt{2}}{2}$

Simplify $\frac{10 \pm 8 \sqrt{2}}{2}$ .

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

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 7$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 7}}{2 \cdot 1}$

Multiply $- 4$ by $- 7$ .

$x = \frac{10 \pm \sqrt{100 + 28}}{2 \cdot 1}$

Add $100$ and $28$ .

$x = \frac{10 \pm \sqrt{128}}{2 \cdot 1}$

Rewrite $128$ as $8^{2} \cdot 2$ .

Factor $64$ out of $128$ .

$x = \frac{10 \pm \sqrt{64 (2)}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{10 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{10 \pm 8 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 8 \sqrt{2}}{2}$

Simplify $\frac{10 \pm 8 \sqrt{2}}{2}$ .

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

Change the $\pm$ to $+$ .

$x = 5 + 4 \sqrt{2}$

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

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 7$ .

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 7}}{2 \cdot 1}$

Multiply $- 4$ by $- 7$ .

$x = \frac{10 \pm \sqrt{100 + 28}}{2 \cdot 1}$

Add $100$ and $28$ .

$x = \frac{10 \pm \sqrt{128}}{2 \cdot 1}$

Rewrite $128$ as $8^{2} \cdot 2$ .

Factor $64$ out of $128$ .

$x = \frac{10 \pm \sqrt{64 (2)}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{10 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{10 \pm 8 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 8 \sqrt{2}}{2}$

Simplify $\frac{10 \pm 8 \sqrt{2}}{2}$ .

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

Change the $\pm$ to $-$ .

$x = 5 - 4 \sqrt{2}$

The final answer is the combination of both solutions.

$x = 5 + 4 \sqrt{2}, 5 - 4 \sqrt{2}$

The final solution is all the values that make $(x + 3) (x^{2} - 10 x - 7) = 0$ true.

$x = - 3, 5 + 4 \sqrt{2}, 5 - 4 \sqrt{2}$

Step 8

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

$(x + 1) (x + 3) (x - 5 - 4 \sqrt{2}) (x - 5 + 4 \sqrt{2})$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 6 x^{3} - 44 x^{2} - 58 x - 21$ .

$x = - 1, - 3, 5 + 4 \sqrt{2}, 5 - 4 \sqrt{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 1, - 3, 5 + 4 \sqrt{2}, 5 - 4 \sqrt{2}$

Decimal Form:

$x = - 1, - 3, 10.65685424 \dots, - 0.65685424 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-6x^3-44x^2-58x-21

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