Find the Roots/Zeros Using the Rational Roots Test 4t^3-18t^2+16t+13

$4 t^{3} - 18 t^{2} + 16 t + 13$

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.

$p = \pm 1, \pm 13$

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

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 13, \pm \frac{13}{2}, \pm \frac{13}{4}$

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.

$4 {(- \frac{1}{2})}^{3} - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $t = - \frac{1}{2}$ is a root of the polynomial.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{1}{2}$ .

$4 ({(- 1)}^{3} {(\frac{1}{2})}^{3}) - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

Apply the product rule to $\frac{1}{2}$ .

$4 ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

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

$4 (- \frac{1^{3}}{2^{3}}) - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

One to any power is one.

$4 (- \frac{1}{2^{3}}) - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

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

$4 (- \frac{1}{8}) - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$4 (\frac{- 1}{8}) - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

Factor $4$ out of $8$ .

$4 \frac{- 1}{4 (2)} - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

Cancel the common factor.

$4 \frac{- 1}{4 \cdot 2} - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

Rewrite the expression.

$\frac{- 1}{2} - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

Move the negative in front of the fraction.

$- \frac{1}{2} - 18 {(- \frac{1}{2})}^{2} + 16 (- \frac{1}{2}) + 13$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{1}{2}$ .

$- \frac{1}{2} - 18 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) + 16 (- \frac{1}{2}) + 13$

Apply the product rule to $\frac{1}{2}$ .

$- \frac{1}{2} - 18 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) + 16 (- \frac{1}{2}) + 13$

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

$- \frac{1}{2} - 18 (1 \frac{1^{2}}{2^{2}}) + 16 (- \frac{1}{2}) + 13$

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

$- \frac{1}{2} - 18 \frac{1^{2}}{2^{2}} + 16 (- \frac{1}{2}) + 13$

One to any power is one.

$- \frac{1}{2} - 18 \frac{1}{2^{2}} + 16 (- \frac{1}{2}) + 13$

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

$- \frac{1}{2} - 18 (\frac{1}{4}) + 16 (- \frac{1}{2}) + 13$

Cancel the common factor of $2$ .

Factor $2$ out of $- 18$ .

$- \frac{1}{2} + 2 (- 9) \frac{1}{4} + 16 (- \frac{1}{2}) + 13$

Factor $2$ out of $4$ .

$- \frac{1}{2} + 2 \cdot - 9 \frac{1}{2 \cdot 2} + 16 (- \frac{1}{2}) + 13$

Cancel the common factor.

$- \frac{1}{2} + 2 \cdot - 9 \frac{1}{2 \cdot 2} + 16 (- \frac{1}{2}) + 13$

Rewrite the expression.

$- \frac{1}{2} - 9 (\frac{1}{2}) + 16 (- \frac{1}{2}) + 13$

Combine $- 9$ and $\frac{1}{2}$ .

$- \frac{1}{2} + \frac{- 9}{2} + 16 (- \frac{1}{2}) + 13$

Move the negative in front of the fraction.

$- \frac{1}{2} - \frac{9}{2} + 16 (- \frac{1}{2}) + 13$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{1}{2} - \frac{9}{2} + 16 (\frac{- 1}{2}) + 13$

Factor $2$ out of $16$ .

$- \frac{1}{2} - \frac{9}{2} + 2 (8) \frac{- 1}{2} + 13$

Cancel the common factor.

$- \frac{1}{2} - \frac{9}{2} + 2 \cdot 8 \frac{- 1}{2} + 13$

Rewrite the expression.

$- \frac{1}{2} - \frac{9}{2} + 8 \cdot - 1 + 13$

Multiply $8$ by $- 1$ .

$- \frac{1}{2} - \frac{9}{2} - 8 + 13$

Combine fractions.

Combine the numerators over the common denominator.

$- 8 + 13 + \frac{- 1 - 9}{2}$

Subtract $9$ from $- 1$ .

$- 8 + 13 + \frac{- 10}{2}$

Find the common denominator.

Write $- 8$ as a fraction with denominator $1$ .

$\frac{- 8}{1} + 13 + \frac{- 10}{2}$

Multiply $\frac{- 8}{1}$ by $\frac{2}{2}$ .

$\frac{- 8}{1} \cdot \frac{2}{2} + 13 + \frac{- 10}{2}$

Multiply $\frac{- 8}{1}$ by $\frac{2}{2}$ .

$\frac{- 8 \cdot 2}{2} + 13 + \frac{- 10}{2}$

Write $13$ as a fraction with denominator $1$ .

$\frac{- 8 \cdot 2}{2} + \frac{13}{1} + \frac{- 10}{2}$

Multiply $\frac{13}{1}$ by $\frac{2}{2}$ .

$\frac{- 8 \cdot 2}{2} + \frac{13}{1} \cdot \frac{2}{2} + \frac{- 10}{2}$

Multiply $\frac{13}{1}$ by $\frac{2}{2}$ .

$\frac{- 8 \cdot 2}{2} + \frac{13 \cdot 2}{2} + \frac{- 10}{2}$

Combine the numerators over the common denominator.

$\frac{- 8 \cdot 2 + 13 \cdot 2 - 10}{2}$

Simplify each term.

Multiply $- 8$ by $2$ .

$\frac{- 16 + 13 \cdot 2 - 10}{2}$

Multiply $13$ by $2$ .

$\frac{- 16 + 26 - 10}{2}$

Simplify the expression.

Add $- 16$ and $26$ .

$\frac{10 - 10}{2}$

Subtract $10$ from $10$ .

$\frac{0}{2}$

Divide $0$ by $2$ .

$0$

Step 5

Since $- \frac{1}{2}$ is a known root, divide the polynomial by $t + \frac{1}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 t^{3} - 18 t^{2} + 16 t + 13}{t + \frac{1}{2}}$

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.

$- \frac{1}{2}$	$4$	$- 18$	$16$	$13$
	\xa0			\xa0

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

$- \frac{1}{2}$	$4$	$- 18$	$16$	$13$
	\xa0			\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 2)$ under the next term in the dividend $(- 18)$ .

$- \frac{1}{2}$	$4$	$- 18$	$16$	$13$
	\xa0	$- 2$		\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.

$- \frac{1}{2}$	$4$	$- 18$	$16$	$13$
	\xa0	$- 2$		\xa0
	$4$	$- 20$

Multiply the newest entry in the result $(- 20)$ by the divisor $(- \frac{1}{2})$ and place the result of $(10)$ under the next term in the dividend $(16)$ .

$- \frac{1}{2}$	$4$	$- 18$	$16$	$13$
	\xa0	$- 2$	$10$	\xa0
	$4$	$- 20$

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

$- \frac{1}{2}$	$4$	$- 18$	$16$	$13$
	\xa0	$- 2$	$10$	\xa0
	$4$	$- 20$	$26$

Multiply the newest entry in the result $(26)$ by the divisor $(- \frac{1}{2})$ and place the result of $(- 13)$ under the next term in the dividend $(13)$ .

$- \frac{1}{2}$	$4$	$- 18$	$16$	$13$
	\xa0	$- 2$	$10$	$- 13$
	$4$	$- 20$	$26$

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

$- \frac{1}{2}$	$4$	$- 18$	$16$	$13$
	\xa0	$- 2$	$10$	$- 13$
	$4$	$- 20$	$26$	$0$

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

$4 t^{2} + - 20 t + 26$

Simplify the quotient polynomial.

$4 t^{2} - 20 t + 26$

Step 7

Factor $2$ out of $4 t^{2} - 20 t + 26$ .

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

$(t + \frac{1}{2}) (2 (2 t^{2}) - 20 t + 26)$

Factor $2$ out of $- 20 t$ .

$(t + \frac{1}{2}) (2 (2 t^{2}) + 2 (- 10 t) + 26)$

Factor $2$ out of $26$ .

$(t + \frac{1}{2}) (2 (2 t^{2}) + 2 (- 10 t) + 2 (13))$

Factor $2$ out of $2 (2 t^{2}) + 2 (- 10 t)$ .

$(t + \frac{1}{2}) (2 (2 t^{2} - 10 t) + 2 (13))$

Factor $2$ out of $2 (2 t^{2} - 10 t) + 2 (13)$ .

$(t + \frac{1}{2}) (2 (2 t^{2} - 10 t + 13))$

Step 8

Factor $4 t^{3} - 18 t^{2} + 16 t + 13$ 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.

$p = \pm 1, \pm 13 q = \pm 1, \pm 4, \pm 2$

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 13, \pm 3.25, \pm 6.5$

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

Substitute $- 0.5$ into the polynomial.

$4 {(- 0.5)}^{3} - 18 {(- 0.5)}^{2} + 16 \cdot - 0.5 + 13$

Raise $- 0.5$ to the power of $3$ .

$4 \cdot - 0.125 - 18 {(- 0.5)}^{2} + 16 \cdot - 0.5 + 13$

Multiply $4$ by $- 0.125$ .

$- 0.5 - 18 {(- 0.5)}^{2} + 16 \cdot - 0.5 + 13$

Raise $- 0.5$ to the power of $2$ .

$- 0.5 - 18 \cdot 0.25 + 16 \cdot - 0.5 + 13$

Multiply $- 18$ by $0.25$ .

$- 0.5 - 4.5 + 16 \cdot - 0.5 + 13$

Subtract $4.5$ from $- 0.5$ .

$- 5 + 16 \cdot - 0.5 + 13$

Multiply $16$ by $- 0.5$ .

$- 5 - 8 + 13$

Subtract $8$ from $- 5$ .

$- 13 + 13$

Add $- 13$ and $13$ .

$0$

Since $- 0.5$ is a known root, divide the polynomial by $2 t + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 t^{3} - 18 t^{2} + 16 t + 13}{2 t + 1}$

Divide $4 t^{3} - 18 t^{2} + 16 t + 13$ by $2 t + 1$ .

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

$2 t$

$1$

$4 t^{3}$

$18 t^{2}$

$16 t$

$13$

Divide the highest order term in the dividend $4 t^{3}$ by the highest order term in divisor $2 t$ .

					$2 t^{2}$
	$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$

Multiply the new quotient term by the divisor.

				$2 t^{2}$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			+	$4 t^{3}$	+	$2 t^{2}$

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

				$2 t^{2}$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$

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

				$2 t^{2}$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$

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

				$2 t^{2}$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$

Divide the highest order term in the dividend $- 20 t^{2}$ by the highest order term in divisor $2 t$ .

				$2 t^{2}$	-	$10 t$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$

Multiply the new quotient term by the divisor.

				$2 t^{2}$	-	$10 t$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$
					-	$20 t^{2}$	-	$10 t$

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 t^{2} - 10 t$

				$2 t^{2}$	-	$10 t$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$
					+	$20 t^{2}$	+	$10 t$

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

				$2 t^{2}$	-	$10 t$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$
					+	$20 t^{2}$	+	$10 t$
							+	$26 t$

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

				$2 t^{2}$	-	$10 t$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$
					+	$20 t^{2}$	+	$10 t$
							+	$26 t$	+	$13$

Divide the highest order term in the dividend $26 t$ by the highest order term in divisor $2 t$ .

				$2 t^{2}$	-	$10 t$	+	$13$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$
					+	$20 t^{2}$	+	$10 t$
							+	$26 t$	+	$13$

Multiply the new quotient term by the divisor.

				$2 t^{2}$	-	$10 t$	+	$13$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$
					+	$20 t^{2}$	+	$10 t$
							+	$26 t$	+	$13$
							+	$26 t$	+	$13$

The expression needs to be subtracted from the dividend, so change all the signs in $26 t + 13$

				$2 t^{2}$	-	$10 t$	+	$13$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$
					+	$20 t^{2}$	+	$10 t$
							+	$26 t$	+	$13$
							-	$26 t$	-	$13$

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

				$2 t^{2}$	-	$10 t$	+	$13$
$2 t$	+	$1$		$4 t^{3}$	-	$18 t^{2}$	+	$16 t$	+	$13$
			-	$4 t^{3}$	-	$2 t^{2}$
					-	$20 t^{2}$	+	$16 t$
					+	$20 t^{2}$	+	$10 t$
							+	$26 t$	+	$13$
							-	$26 t$	-	$13$
										$0$

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

$2 t^{2} - 10 t + 13$

Write $4 t^{3} - 18 t^{2} + 16 t + 13$ as a set of factors.

$(2 t + 1) (2 t^{2} - 10 t + 13) = 0$

Step 9

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

$2 t + 1 = 0$

$2 t^{2} - 10 t + 13 = 0$

Step 10

Set $2 t + 1$ equal to $0$ and solve for $t$ .

Set $2 t + 1$ equal to $0$ .

$2 t + 1 = 0$

Solve $2 t + 1 = 0$ for $t$ .

Subtract $1$ from both sides of the equation.

$2 t = - 1$

Divide each term in $2 t = - 1$ by $2$ and simplify.

Divide each term in $2 t = - 1$ by $2$ .

$\frac{2 t}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 t}{2} = \frac{- 1}{2}$

Divide $t$ by $1$ .

$t = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$t = - \frac{1}{2}$

Step 11

Set $2 t^{2} - 10 t + 13$ equal to $0$ and solve for $t$ .

Set $2 t^{2} - 10 t + 13$ equal to $0$ .

$2 t^{2} - 10 t + 13 = 0$

Solve $2 t^{2} - 10 t + 13 = 0$ for $t$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 2$ , $b = - 10$ , and $c = 13$ into the quadratic formula and solve for $t$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (2 \cdot 13)}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

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

$t = \frac{10 \pm \sqrt{100 - 4 \cdot 2 \cdot 13}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 13$ .

Multiply $- 4$ by $2$ .

$t = \frac{10 \pm \sqrt{100 - 8 \cdot 13}}{2 \cdot 2}$

Multiply $- 8$ by $13$ .

$t = \frac{10 \pm \sqrt{100 - 104}}{2 \cdot 2}$

Subtract $104$ from $100$ .

$t = \frac{10 \pm \sqrt{- 4}}{2 \cdot 2}$

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

$t = \frac{10 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 2}$

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

$t = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 2}$

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

$t = \frac{10 \pm i \cdot \sqrt{4}}{2 \cdot 2}$

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

$t = \frac{10 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 2}$

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

$t = \frac{10 \pm i \cdot 2}{2 \cdot 2}$

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

$t = \frac{10 \pm 2 i}{2 \cdot 2}$

Multiply $2$ by $2$ .

$t = \frac{10 \pm 2 i}{4}$

Simplify $\frac{10 \pm 2 i}{4}$ .

$t = \frac{5 \pm i}{2}$

The final answer is the combination of both solutions.

$t = \frac{5}{2} + \frac{i}{2}, \frac{5}{2} - \frac{i}{2}$

Step 12

The final solution is all the values that make $(2 t + 1) (2 t^{2} - 10 t + 13) = 0$ true.

$t = - \frac{1}{2}, \frac{5}{2} + \frac{i}{2}, \frac{5}{2} - \frac{i}{2}$

Step 13

Find the Roots/Zeros Using the Rational Roots Test 4t^3-18t^2+16t+13

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