Find the Roots/Zeros Using the Rational Roots Test f(x)=2x^4-19x^3+71x^2-109x+39

$f (x) = 2 x^{4} - 19 x^{3} + 71 x^{2} - 109 x + 39$

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.

$p = \pm 1, \pm 3, \pm 13, \pm 39$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 13, \pm \frac{13}{2}, \pm 39, \pm \frac{39}{2}$

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.

$2 {(\frac{1}{2})}^{4} - 19 {(\frac{1}{2})}^{3} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

Step 4

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

Simplify each term.

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

$2 \frac{1^{4}}{2^{4}} - 19 {(\frac{1}{2})}^{3} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

One to any power is one.

$2 \frac{1}{2^{4}} - 19 {(\frac{1}{2})}^{3} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

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

$2 (\frac{1}{16}) - 19 {(\frac{1}{2})}^{3} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$2 \frac{1}{2 (8)} - 19 {(\frac{1}{2})}^{3} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

Cancel the common factor.

$2 \frac{1}{2 \cdot 8} - 19 {(\frac{1}{2})}^{3} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

Rewrite the expression.

$\frac{1}{8} - 19 {(\frac{1}{2})}^{3} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

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

$\frac{1}{8} - 19 \frac{1^{3}}{2^{3}} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

One to any power is one.

$\frac{1}{8} - 19 \frac{1}{2^{3}} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

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

$\frac{1}{8} - 19 (\frac{1}{8}) + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

Combine $- 19$ and $\frac{1}{8}$ .

$\frac{1}{8} + \frac{- 19}{8} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

Move the negative in front of the fraction.

$\frac{1}{8} - \frac{19}{8} + 71 {(\frac{1}{2})}^{2} - 109 (\frac{1}{2}) + 39$

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

$\frac{1}{8} - \frac{19}{8} + 71 \frac{1^{2}}{2^{2}} - 109 (\frac{1}{2}) + 39$

One to any power is one.

$\frac{1}{8} - \frac{19}{8} + 71 \frac{1}{2^{2}} - 109 (\frac{1}{2}) + 39$

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

$\frac{1}{8} - \frac{19}{8} + 71 (\frac{1}{4}) - 109 (\frac{1}{2}) + 39$

Combine $71$ and $\frac{1}{4}$ .

$\frac{1}{8} - \frac{19}{8} + \frac{71}{4} - 109 (\frac{1}{2}) + 39$

Combine $- 109$ and $\frac{1}{2}$ .

$\frac{1}{8} - \frac{19}{8} + \frac{71}{4} + \frac{- 109}{2} + 39$

Move the negative in front of the fraction.

$\frac{1}{8} - \frac{19}{8} + \frac{71}{4} - \frac{109}{2} + 39$

Combine fractions.

Combine the numerators over the common denominator.

$39 + \frac{1 - 19}{8} + \frac{71}{4} + \frac{- 109}{2}$

Subtract $19$ from $1$ .

$39 + \frac{- 18}{8} + \frac{71}{4} + \frac{- 109}{2}$

Find the common denominator.

Write $39$ as a fraction with denominator $1$ .

$\frac{39}{1} + \frac{- 18}{8} + \frac{71}{4} + \frac{- 109}{2}$

Multiply $\frac{39}{1}$ by $\frac{8}{8}$ .

$\frac{39}{1} \cdot \frac{8}{8} + \frac{- 18}{8} + \frac{71}{4} + \frac{- 109}{2}$

Multiply $\frac{39}{1}$ by $\frac{8}{8}$ .

$\frac{39 \cdot 8}{8} + \frac{- 18}{8} + \frac{71}{4} + \frac{- 109}{2}$

Multiply $\frac{71}{4}$ by $\frac{2}{2}$ .

$\frac{39 \cdot 8}{8} + \frac{- 18}{8} + \frac{71}{4} \cdot \frac{2}{2} + \frac{- 109}{2}$

Multiply $\frac{71}{4}$ by $\frac{2}{2}$ .

$\frac{39 \cdot 8}{8} + \frac{- 18}{8} + \frac{71 \cdot 2}{4 \cdot 2} + \frac{- 109}{2}$

Multiply $\frac{- 109}{2}$ by $\frac{4}{4}$ .

$\frac{39 \cdot 8}{8} + \frac{- 18}{8} + \frac{71 \cdot 2}{4 \cdot 2} + \frac{- 109}{2} \cdot \frac{4}{4}$

Multiply $\frac{- 109}{2}$ by $\frac{4}{4}$ .

$\frac{39 \cdot 8}{8} + \frac{- 18}{8} + \frac{71 \cdot 2}{4 \cdot 2} + \frac{- 109 \cdot 4}{2 \cdot 4}$

Reorder the factors of $4 \cdot 2$ .

$\frac{39 \cdot 8}{8} + \frac{- 18}{8} + \frac{71 \cdot 2}{2 \cdot 4} + \frac{- 109 \cdot 4}{2 \cdot 4}$

Multiply $2$ by $4$ .

$\frac{39 \cdot 8}{8} + \frac{- 18}{8} + \frac{71 \cdot 2}{8} + \frac{- 109 \cdot 4}{2 \cdot 4}$

Multiply $2$ by $4$ .

$\frac{39 \cdot 8}{8} + \frac{- 18}{8} + \frac{71 \cdot 2}{8} + \frac{- 109 \cdot 4}{8}$

Combine the numerators over the common denominator.

$\frac{39 \cdot 8 - 18 + 71 \cdot 2 - 109 \cdot 4}{8}$

Simplify each term.

Multiply $39$ by $8$ .

$\frac{312 - 18 + 71 \cdot 2 - 109 \cdot 4}{8}$

Multiply $71$ by $2$ .

$\frac{312 - 18 + 142 - 109 \cdot 4}{8}$

Multiply $- 109$ by $4$ .

$\frac{312 - 18 + 142 - 436}{8}$

Simplify the expression.

Subtract $18$ from $312$ .

$\frac{294 + 142 - 436}{8}$

Add $294$ and $142$ .

$\frac{436 - 436}{8}$

Subtract $436$ from $436$ .

$\frac{0}{8}$

Divide $0$ by $8$ .

$0$

Step 5

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

$\frac{2 x^{4} - 19 x^{3} + 71 x^{2} - 109 x + 39}{x - \frac{1}{2}}$

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.

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0				\xa0

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0				\xa0
	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(\frac{1}{2})$ and place the result of $(1)$ under the next term in the dividend $(- 19)$ .

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0	$1$			\xa0
	$2$

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

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0	$1$			\xa0
	$2$	$- 18$

Multiply the newest entry in the result $(- 18)$ by the divisor $(\frac{1}{2})$ and place the result of $(- 9)$ under the next term in the dividend $(71)$ .

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0	$1$	$- 9$		\xa0
	$2$	$- 18$

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

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0	$1$	$- 9$		\xa0
	$2$	$- 18$	$62$

Multiply the newest entry in the result $(62)$ by the divisor $(\frac{1}{2})$ and place the result of $(31)$ under the next term in the dividend $(- 109)$ .

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0	$1$	$- 9$	$31$	\xa0
	$2$	$- 18$	$62$

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

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0	$1$	$- 9$	$31$	\xa0
	$2$	$- 18$	$62$	$- 78$

Multiply the newest entry in the result $(- 78)$ by the divisor $(\frac{1}{2})$ and place the result of $(- 39)$ under the next term in the dividend $(39)$ .

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0	$1$	$- 9$	$31$	$- 39$
	$2$	$- 18$	$62$	$- 78$

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

$\frac{1}{2}$	$2$	$- 19$	$71$	$- 109$	$39$
	\xa0	$1$	$- 9$	$31$	$- 39$
	$2$	$- 18$	$62$	$- 78$	$0$

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

$2 x^{3} + - 18 x^{2} + (62) x - 78$

Simplify the quotient polynomial.

$2 x^{3} - 18 x^{2} + 62 x - 78$

Step 7

Factor $2$ out of $2 x^{3} - 18 x^{2} + 62 x - 78$ .

Factor $2$ out of $2 x^{3}$ .

$(x - \frac{1}{2}) (2 (x^{3}) - 18 x^{2} + 62 x - 78)$

Factor $2$ out of $- 18 x^{2}$ .

$(x - \frac{1}{2}) (2 (x^{3}) + 2 (- 9 x^{2}) + 62 x - 78)$

Factor $2$ out of $62 x$ .

$(x - \frac{1}{2}) (2 (x^{3}) + 2 (- 9 x^{2}) + 2 (31 x) - 78)$

Factor $2$ out of $- 78$ .

$(x - \frac{1}{2}) (2 x^{3} + 2 (- 9 x^{2}) + 2 (31 x) + 2 \cdot - 39)$

Factor $2$ out of $2 x^{3} + 2 (- 9 x^{2})$ .

$(x - \frac{1}{2}) (2 (x^{3} - 9 x^{2}) + 2 (31 x) + 2 \cdot - 39)$

Factor $2$ out of $2 (x^{3} - 9 x^{2}) + 2 (31 x)$ .

$(x - \frac{1}{2}) (2 (x^{3} - 9 x^{2} + 31 x) + 2 \cdot - 39)$

Factor $2$ out of $2 (x^{3} - 9 x^{2} + 31 x) + 2 \cdot - 39$ .

$(x - \frac{1}{2}) (2 (x^{3} - 9 x^{2} + 31 x - 39))$

Step 8

Factor the left side of the equation.

Factor $2 x^{4} - 19 x^{3} + 71 x^{2} - 109 x + 39$ 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.

$p = \pm 1, \pm 39, \pm 3, \pm 13 q = \pm 1, \pm 2$

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

$\pm 1, \pm 0.5, \pm 39, \pm 19.5, \pm 3, \pm 1.5, \pm 13, \pm 6.5$

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.

$2 \cdot {0.5}^{4} - 19 \cdot {0.5}^{3} + 71 \cdot {0.5}^{2} - 109 \cdot 0.5 + 39$

Raise $0.5$ to the power of $4$ .

$2 \cdot 0.0625 - 19 \cdot {0.5}^{3} + 71 \cdot {0.5}^{2} - 109 \cdot 0.5 + 39$

Multiply $2$ by $0.0625$ .

$0.125 - 19 \cdot {0.5}^{3} + 71 \cdot {0.5}^{2} - 109 \cdot 0.5 + 39$

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

$0.125 - 19 \cdot 0.125 + 71 \cdot {0.5}^{2} - 109 \cdot 0.5 + 39$

Multiply $- 19$ by $0.125$ .

$0.125 - 2.375 + 71 \cdot {0.5}^{2} - 109 \cdot 0.5 + 39$

Subtract $2.375$ from $0.125$ .

$- 2.25 + 71 \cdot {0.5}^{2} - 109 \cdot 0.5 + 39$

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

$- 2.25 + 71 \cdot 0.25 - 109 \cdot 0.5 + 39$

Multiply $71$ by $0.25$ .

$- 2.25 + 17.75 - 109 \cdot 0.5 + 39$

Add $- 2.25$ and $17.75$ .

$15.5 - 109 \cdot 0.5 + 39$

Multiply $- 109$ by $0.5$ .

$15.5 - 54.5 + 39$

Subtract $54.5$ from $15.5$ .

$- 39 + 39$

Add $- 39$ and $39$ .

$0$

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

$\frac{2 x^{4} - 19 x^{3} + 71 x^{2} - 109 x + 39}{2 x - 1}$

Divide $2 x^{4} - 19 x^{3} + 71 x^{2} - 109 x + 39$ by $2 x - 1$ .

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

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

Divide the highest order term in the dividend $- 18 x^{3}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$9 x^{2}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$9 x^{2}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{3} + 9 x^{2}$

$x^{3}$

$9 x^{2}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

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

$x^{3}$

$9 x^{2}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

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

$x^{3}$

$9 x^{2}$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

Divide the highest order term in the dividend $62 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$9 x^{2}$

$31 x$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$9 x^{2}$

$31 x$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

$62 x^{2}$

$31 x$

The expression needs to be subtracted from the dividend, so change all the signs in $62 x^{2} - 31 x$

$x^{3}$

$9 x^{2}$

$31 x$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

$62 x^{2}$

$31 x$

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

$x^{3}$

$9 x^{2}$

$31 x$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

$62 x^{2}$

$31 x$

$78 x$

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

$x^{3}$

$9 x^{2}$

$31 x$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

$62 x^{2}$

$31 x$

$78 x$

$39$

Divide the highest order term in the dividend $- 78 x$ by the highest order term in divisor $2 x$ .

$x^{3}$

$9 x^{2}$

$31 x$

$39$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

$62 x^{2}$

$31 x$

$78 x$

$39$

Multiply the new quotient term by the divisor.

$x^{3}$

$9 x^{2}$

$31 x$

$39$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

$62 x^{2}$

$31 x$

$78 x$

$39$

$78 x$

$39$

The expression needs to be subtracted from the dividend, so change all the signs in $- 78 x + 39$

$x^{3}$

$9 x^{2}$

$31 x$

$39$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

$62 x^{2}$

$31 x$

$78 x$

$39$

$78 x$

$39$

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

$x^{3}$

$9 x^{2}$

$31 x$

$39$

$2 x$

$1$

$2 x^{4}$

$19 x^{3}$

$71 x^{2}$

$109 x$

$39$

$2 x^{4}$

$x^{3}$

$18 x^{3}$

$71 x^{2}$

$18 x^{3}$

$9 x^{2}$

$62 x^{2}$

$109 x$

$62 x^{2}$

$31 x$

$78 x$

$39$

$78 x$

$39$

$0$

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

$x^{3} - 9 x^{2} + 31 x - 39$

Write $2 x^{4} - 19 x^{3} + 71 x^{2} - 109 x + 39$ as a set of factors.

$(2 x - 1) (x^{3} - 9 x^{2} + 31 x - 39) = 0$

Factor $x^{3} - 9 x^{2} + 31 x - 39$ 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.

$p = \pm 1, \pm 39, \pm 3, \pm 13 q = \pm 1$

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

$\pm 1, \pm 39, \pm 3, \pm 13$

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.

$3^{3} - 9 \cdot 3^{2} + 31 \cdot 3 - 39$

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

$27 - 9 \cdot 3^{2} + 31 \cdot 3 - 39$

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

$27 - 9 \cdot 9 + 31 \cdot 3 - 39$

Multiply $- 9$ by $9$ .

$27 - 81 + 31 \cdot 3 - 39$

Subtract $81$ from $27$ .

$- 54 + 31 \cdot 3 - 39$

Multiply $31$ by $3$ .

$- 54 + 93 - 39$

Add $- 54$ and $93$ .

$39 - 39$

Subtract $39$ from $39$ .

$0$

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.

$\frac{x^{3} - 9 x^{2} + 31 x - 39}{x - 3}$

Divide $x^{3} - 9 x^{2} + 31 x - 39$ 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 $0$ .

$x$

$3$

$x^{3}$

$9 x^{2}$

$31 x$

$39$

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			+	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$
					-	$6 x^{2}$	+	$18 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 18 x$

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$
					+	$6 x^{2}$	-	$18 x$

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

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$
					+	$6 x^{2}$	-	$18 x$
							+	$13 x$

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

				$x^{2}$	-	$6 x$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$
					+	$6 x^{2}$	-	$18 x$
							+	$13 x$	-	$39$

Divide the highest order term in the dividend $13 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$	+	$13$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$
					+	$6 x^{2}$	-	$18 x$
							+	$13 x$	-	$39$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	+	$13$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$
					+	$6 x^{2}$	-	$18 x$
							+	$13 x$	-	$39$
							+	$13 x$	-	$39$

The expression needs to be subtracted from the dividend, so change all the signs in $13 x - 39$

				$x^{2}$	-	$6 x$	+	$13$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$
					+	$6 x^{2}$	-	$18 x$
							+	$13 x$	-	$39$
							-	$13 x$	+	$39$

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

				$x^{2}$	-	$6 x$	+	$13$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$31 x$	-	$39$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$31 x$
					+	$6 x^{2}$	-	$18 x$
							+	$13 x$	-	$39$
							-	$13 x$	+	$39$
										$0$

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

$x^{2} - 6 x + 13$

Write $x^{3} - 9 x^{2} + 31 x - 39$ as a set of factors.

$(2 x - 1) ((x - 3) (x^{2} - 6 x + 13)) = 0$

Remove unnecessary parentheses.

$(2 x - 1) (x - 3) (x^{2} - 6 x + 13) = 0$

Step 9

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

$2 x - 1 = 0$

$x - 3 = 0$

$x^{2} - 6 x + 13 = 0$

Step 10

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

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

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 11

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 12

Set $x^{2} - 6 x + 13$ equal to $0$ and solve for $x$ .

Set $x^{2} - 6 x + 13$ equal to $0$ .

$x^{2} - 6 x + 13 = 0$

Solve $x^{2} - 6 x + 13 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 6$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 13$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 13}}{2 \cdot 1}$

Multiply $- 4$ by $13$ .

$x = \frac{6 \pm \sqrt{36 - 52}}{2 \cdot 1}$

Subtract $52$ from $36$ .

$x = \frac{6 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{6 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 4 i}{2}$

Simplify $\frac{6 \pm 4 i}{2}$ .

$x = 3 \pm 2 i$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 13$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 13}}{2 \cdot 1}$

Multiply $- 4$ by $13$ .

$x = \frac{6 \pm \sqrt{36 - 52}}{2 \cdot 1}$

Subtract $52$ from $36$ .

$x = \frac{6 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{6 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 4 i}{2}$

Simplify $\frac{6 \pm 4 i}{2}$ .

$x = 3 \pm 2 i$

Change the $\pm$ to $+$ .

$x = 3 + 2 i$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 13$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 13}}{2 \cdot 1}$

Multiply $- 4$ by $13$ .

$x = \frac{6 \pm \sqrt{36 - 52}}{2 \cdot 1}$

Subtract $52$ from $36$ .

$x = \frac{6 \pm \sqrt{- 16}}{2 \cdot 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot \sqrt{16}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot 4}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{6 \pm 4 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 4 i}{2}$

Simplify $\frac{6 \pm 4 i}{2}$ .

$x = 3 \pm 2 i$

Change the $\pm$ to $-$ .

$x = 3 - 2 i$

The final answer is the combination of both solutions.

$x = 3 + 2 i, 3 - 2 i$

Step 13

The final solution is all the values that make $(2 x - 1) (x - 3) (x^{2} - 6 x + 13) = 0$ true.

$x = \frac{1}{2}, 3, 3 + 2 i, 3 - 2 i$

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=2x^4-19x^3+71x^2-109x+39

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