Find the Roots/Zeros Using the Rational Roots Test f(x)=4x^3-13x^2-37x+10

$f (x) = 4 x^{3} - 13 x^{2} - 37 x + 10$

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.

$p = \pm 1, \pm 2, \pm 5, \pm 10$

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

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}, \pm 10$

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.

$4 {(\frac{1}{4})}^{3} - 13 {(\frac{1}{4})}^{2} - 37 (\frac{1}{4}) + 10$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = \frac{1}{4}$ is a root of the polynomial.

Simplify each term.

Apply the product rule to $\frac{1}{4}$ .

$4 \frac{1^{3}}{4^{3}} - 13 {(\frac{1}{4})}^{2} - 37 (\frac{1}{4}) + 10$

One to any power is one.

$4 \frac{1}{4^{3}} - 13 {(\frac{1}{4})}^{2} - 37 (\frac{1}{4}) + 10$

Raise $4$ to the power of $3$ .

$4 (\frac{1}{64}) - 13 {(\frac{1}{4})}^{2} - 37 (\frac{1}{4}) + 10$

Cancel the common factor of $4$ .

Factor $4$ out of $64$ .

$4 \frac{1}{4 (16)} - 13 {(\frac{1}{4})}^{2} - 37 (\frac{1}{4}) + 10$

Cancel the common factor.

$4 \frac{1}{4 \cdot 16} - 13 {(\frac{1}{4})}^{2} - 37 (\frac{1}{4}) + 10$

Rewrite the expression.

$\frac{1}{16} - 13 {(\frac{1}{4})}^{2} - 37 (\frac{1}{4}) + 10$

Apply the product rule to $\frac{1}{4}$ .

$\frac{1}{16} - 13 \frac{1^{2}}{4^{2}} - 37 (\frac{1}{4}) + 10$

One to any power is one.

$\frac{1}{16} - 13 \frac{1}{4^{2}} - 37 (\frac{1}{4}) + 10$

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

$\frac{1}{16} - 13 (\frac{1}{16}) - 37 (\frac{1}{4}) + 10$

Combine $- 13$ and $\frac{1}{16}$ .

$\frac{1}{16} + \frac{- 13}{16} - 37 (\frac{1}{4}) + 10$

Move the negative in front of the fraction.

$\frac{1}{16} - \frac{13}{16} - 37 (\frac{1}{4}) + 10$

Combine $- 37$ and $\frac{1}{4}$ .

$\frac{1}{16} - \frac{13}{16} + \frac{- 37}{4} + 10$

Move the negative in front of the fraction.

$\frac{1}{16} - \frac{13}{16} - \frac{37}{4} + 10$

Combine fractions.

Combine the numerators over the common denominator.

$10 + \frac{1 - 13}{16} + \frac{- 37}{4}$

Subtract $13$ from $1$ .

$10 + \frac{- 12}{16} + \frac{- 37}{4}$

Find the common denominator.

Write $10$ as a fraction with denominator $1$ .

$\frac{10}{1} + \frac{- 12}{16} + \frac{- 37}{4}$

Multiply $\frac{10}{1}$ by $\frac{16}{16}$ .

$\frac{10}{1} \cdot \frac{16}{16} + \frac{- 12}{16} + \frac{- 37}{4}$

Multiply $\frac{10}{1}$ by $\frac{16}{16}$ .

$\frac{10 \cdot 16}{16} + \frac{- 12}{16} + \frac{- 37}{4}$

Multiply $\frac{- 37}{4}$ by $\frac{4}{4}$ .

$\frac{10 \cdot 16}{16} + \frac{- 12}{16} + \frac{- 37}{4} \cdot \frac{4}{4}$

Multiply $\frac{- 37}{4}$ by $\frac{4}{4}$ .

$\frac{10 \cdot 16}{16} + \frac{- 12}{16} + \frac{- 37 \cdot 4}{4 \cdot 4}$

Multiply $4$ by $4$ .

$\frac{10 \cdot 16}{16} + \frac{- 12}{16} + \frac{- 37 \cdot 4}{16}$

Combine the numerators over the common denominator.

$\frac{10 \cdot 16 - 12 - 37 \cdot 4}{16}$

Simplify each term.

Multiply $10$ by $16$ .

$\frac{160 - 12 - 37 \cdot 4}{16}$

Multiply $- 37$ by $4$ .

$\frac{160 - 12 - 148}{16}$

Simplify the expression.

Subtract $12$ from $160$ .

$\frac{148 - 148}{16}$

Subtract $148$ from $148$ .

$\frac{0}{16}$

Divide $0$ by $16$ .

$0$

Step 5

Since $\frac{1}{4}$ is a known root, divide the polynomial by $x - \frac{1}{4}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 13 x^{2} - 37 x + 10}{x - \frac{1}{4}}$

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.

$\frac{1}{4}$	$4$	$- 13$	$- 37$	$10$
	\xa0			\xa0

The first number in the dividend $(4)$ is put into the first position of the result area (below the horizontal line).

$\frac{1}{4}$	$4$	$- 13$	$- 37$	$10$
	\xa0			\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{1}{4})$ and place the result of $(1)$ under the next term in the dividend $(- 13)$ .

$\frac{1}{4}$	$4$	$- 13$	$- 37$	$10$
	\xa0	$1$		\xa0
	$4$

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

$\frac{1}{4}$	$4$	$- 13$	$- 37$	$10$
	\xa0	$1$		\xa0
	$4$	$- 12$

Multiply the newest entry in the result $(- 12)$ by the divisor $(\frac{1}{4})$ and place the result of $(- 3)$ under the next term in the dividend $(- 37)$ .

$\frac{1}{4}$	$4$	$- 13$	$- 37$	$10$
	\xa0	$1$	$- 3$	\xa0
	$4$	$- 12$

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

$\frac{1}{4}$	$4$	$- 13$	$- 37$	$10$
	\xa0	$1$	$- 3$	\xa0
	$4$	$- 12$	$- 40$

Multiply the newest entry in the result $(- 40)$ by the divisor $(\frac{1}{4})$ and place the result of $(- 10)$ under the next term in the dividend $(10)$ .

$\frac{1}{4}$	$4$	$- 13$	$- 37$	$10$
	\xa0	$1$	$- 3$	$- 10$
	$4$	$- 12$	$- 40$

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

$\frac{1}{4}$	$4$	$- 13$	$- 37$	$10$
	\xa0	$1$	$- 3$	$- 10$
	$4$	$- 12$	$- 40$	$0$

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

$4 x^{2} + - 12 x - 40$

Simplify the quotient polynomial.

$4 x^{2} - 12 x - 40$

Step 7

Factor $4$ out of $4 x^{2} - 12 x - 40$ .

Factor $4$ out of $4 x^{2}$ .

$(x - \frac{1}{4}) (4 (x^{2}) - 12 x - 40)$

Factor $4$ out of $- 12 x$ .

$(x - \frac{1}{4}) (4 (x^{2}) + 4 (- 3 x) - 40)$

Factor $4$ out of $- 40$ .

$(x - \frac{1}{4}) (4 x^{2} + 4 (- 3 x) + 4 \cdot - 10)$

Factor $4$ out of $4 x^{2} + 4 (- 3 x)$ .

$(x - \frac{1}{4}) (4 (x^{2} - 3 x) + 4 \cdot - 10)$

Factor $4$ out of $4 (x^{2} - 3 x) + 4 \cdot - 10$ .

$(x - \frac{1}{4}) (4 (x^{2} - 3 x - 10))$

Step 8

Factor.

Factor $x^{2} - 3 x - 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Step 9

Factor the left side of the equation.

Factor $4 x^{3} - 13 x^{2} - 37 x + 10$ 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.

$p = \pm 1, \pm 10, \pm 2, \pm 5 q = \pm 1, \pm 4, \pm 2$

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 10, \pm 2.5, \pm 5, \pm 2, \pm 1.25$

Substitute $0.25$ and simplify the expression. In this case, the expression is equal to $0$ so $0.25$ is a root of the polynomial.

Substitute $0.25$ into the polynomial.

$4 \cdot {0.25}^{3} - 13 \cdot {0.25}^{2} - 37 \cdot 0.25 + 10$

Raise $0.25$ to the power of $3$ .

$4 \cdot 0.015625 - 13 \cdot {0.25}^{2} - 37 \cdot 0.25 + 10$

Multiply $4$ by $0.015625$ .

$0.0625 - 13 \cdot {0.25}^{2} - 37 \cdot 0.25 + 10$

Raise $0.25$ to the power of $2$ .

$0.0625 - 13 \cdot 0.0625 - 37 \cdot 0.25 + 10$

Multiply $- 13$ by $0.0625$ .

$0.0625 - 0.8125 - 37 \cdot 0.25 + 10$

Subtract $0.8125$ from $0.0625$ .

$- 0.75 - 37 \cdot 0.25 + 10$

Multiply $- 37$ by $0.25$ .

$- 0.75 - 9.25 + 10$

Subtract $9.25$ from $- 0.75$ .

$- 10 + 10$

Add $- 10$ and $10$ .

$0$

Since $0.25$ is a known root, divide the polynomial by $4 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 13 x^{2} - 37 x + 10}{4 x - 1}$

Divide $4 x^{3} - 13 x^{2} - 37 x + 10$ by $4 x - 1$ .

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

$4 x$

$1$

$4 x^{3}$

$13 x^{2}$

$37 x$

$10$

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $4 x$ .

					$x^{2}$
	$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			+	$4 x^{3}$	-	$x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - x^{2}$

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$

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

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$

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

				$x^{2}$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$3 x$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$
					-	$12 x^{2}$	+	$3 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} + 3 x$

				$x^{2}$	-	$3 x$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$
					+	$12 x^{2}$	-	$3 x$

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

				$x^{2}$	-	$3 x$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$
					+	$12 x^{2}$	-	$3 x$
							-	$40 x$

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

				$x^{2}$	-	$3 x$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$
					+	$12 x^{2}$	-	$3 x$
							-	$40 x$	+	$10$

Divide the highest order term in the dividend $- 40 x$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$3 x$	-	$10$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$
					+	$12 x^{2}$	-	$3 x$
							-	$40 x$	+	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$	-	$10$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$
					+	$12 x^{2}$	-	$3 x$
							-	$40 x$	+	$10$
							-	$40 x$	+	$10$

The expression needs to be subtracted from the dividend, so change all the signs in $- 40 x + 10$

				$x^{2}$	-	$3 x$	-	$10$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$
					+	$12 x^{2}$	-	$3 x$
							-	$40 x$	+	$10$
							+	$40 x$	-	$10$

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

				$x^{2}$	-	$3 x$	-	$10$
$4 x$	-	$1$		$4 x^{3}$	-	$13 x^{2}$	-	$37 x$	+	$10$
			-	$4 x^{3}$	+	$x^{2}$
					-	$12 x^{2}$	-	$37 x$
					+	$12 x^{2}$	-	$3 x$
							-	$40 x$	+	$10$
							+	$40 x$	-	$10$
										$0$

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

$x^{2} - 3 x - 10$

Write $4 x^{3} - 13 x^{2} - 37 x + 10$ as a set of factors.

$(4 x - 1) (x^{2} - 3 x - 10) = 0$

Factor $x^{2} - 3 x - 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

$(4 x - 1) ((x - 5) (x + 2)) = 0$

Remove unnecessary parentheses.

$(4 x - 1) (x - 5) (x + 2) = 0$

Step 10

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

$4 x - 1 = 0$

$x - 5 = 0$

$x + 2 = 0$

Step 11

Set $4 x - 1$ equal to $0$ and solve for $x$ .

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Solve $4 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 12

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 13

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.

$x = - 2$

Step 14

The final solution is all the values that make $(4 x - 1) (x - 5) (x + 2) = 0$ true.

$x = \frac{1}{4}, 5, - 2$

Step 15

Find the Roots/Zeros Using the Rational Roots Test f(x)=4x^3-13x^2-37x+10

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