Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3-11x^2+43x-65

$f (x) = x^{3} - 11 x^{2} + 43 x - 65$

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.

$p = \pm 1, \pm 5, \pm 13, \pm 65$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 5, \pm 13, \pm 65$

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.

${(5)}^{3} - 11 {(5)}^{2} + 43 (5) - 65$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 5$ is a root of the polynomial.

Simplify each term.

Raise $5$ to the power of $3$ .

$125 - 11 {(5)}^{2} + 43 (5) - 65$

Raise $5$ to the power of $2$ .

$125 - 11 \cdot 25 + 43 (5) - 65$

Multiply $- 11$ by $25$ .

$125 - 275 + 43 (5) - 65$

Multiply $43$ by $5$ .

$125 - 275 + 215 - 65$

Simplify by adding and subtracting.

Subtract $275$ from $125$ .

$- 150 + 215 - 65$

Add $- 150$ and $215$ .

$65 - 65$

Subtract $65$ from $65$ .

$0$

Step 5

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 11 x^{2} + 43 x - 65}{x - 5}$

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.

$5$	$1$	$- 11$	$43$	$- 65$
	\xa0			\xa0

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

$5$	$1$	$- 11$	$43$	$- 65$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(5)$ and place the result of $(5)$ under the next term in the dividend $(- 11)$ .

$5$	$1$	$- 11$	$43$	$- 65$
	\xa0	$5$		\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.

$5$	$1$	$- 11$	$43$	$- 65$
	\xa0	$5$		\xa0
	$1$	$- 6$

Multiply the newest entry in the result $(- 6)$ by the divisor $(5)$ and place the result of $(- 30)$ under the next term in the dividend $(43)$ .

$5$	$1$	$- 11$	$43$	$- 65$
	\xa0	$5$	$- 30$	\xa0
	$1$	$- 6$

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

$5$	$1$	$- 11$	$43$	$- 65$
	\xa0	$5$	$- 30$	\xa0
	$1$	$- 6$	$13$

Multiply the newest entry in the result $(13)$ by the divisor $(5)$ and place the result of $(65)$ under the next term in the dividend $(- 65)$ .

$5$	$1$	$- 11$	$43$	$- 65$
	\xa0	$5$	$- 30$	$65$
	$1$	$- 6$	$13$

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

$5$	$1$	$- 11$	$43$	$- 65$
	\xa0	$5$	$- 30$	$65$
	$1$	$- 6$	$13$	$0$

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

$1 x^{2} + - 6 x + 13$

Simplify the quotient polynomial.

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

Step 7

Solve the equation to find any remaining roots.

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 8

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

$(x - 5) (x - 3 - 2 i) (x - 3 + 2 i)$

Step 9

These are the roots (zeros) of the polynomial $x^{3} - 11 x^{2} + 43 x - 65$ .

$x = 5, 3 + 2 i, 3 - 2 i$

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3-11x^2+43x-65

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