Find the Roots/Zeros Using the Rational Roots Test f(x)=2x^3-4x^2-22x+44

$f (x) = 2 x^{3} - 4 x^{2} - 22 x + 44$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 11, \pm 22, \pm 44$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 2, \pm 4, \pm 11, \pm \frac{11}{2}, \pm 22, \pm 44$

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.

$2 {(2)}^{3} - 4 {(2)}^{2} - 22 \cdot 2 + 44$

Step 4

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

Simplify each term.

Multiply $2$ by ${(2)}^{3}$ by adding the exponents.

Multiply $2$ by ${(2)}^{3}$ .

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

$2^{1} {(2)}^{3} - 4 {(2)}^{2} - 22 \cdot 2 + 44$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + 3} - 4 {(2)}^{2} - 22 \cdot 2 + 44$

Add $1$ and $3$ .

$2^{4} - 4 {(2)}^{2} - 22 \cdot 2 + 44$

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

$16 - 4 {(2)}^{2} - 22 \cdot 2 + 44$

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

$16 - 4 \cdot 4 - 22 \cdot 2 + 44$

Multiply $- 4$ by $4$ .

$16 - 16 - 22 \cdot 2 + 44$

Multiply $- 22$ by $2$ .

$16 - 16 - 44 + 44$

Simplify by adding and subtracting.

Subtract $16$ from $16$ .

$0 - 44 + 44$

Subtract $44$ from $0$ .

$- 44 + 44$

Add $- 44$ and $44$ .

$0$

Step 5

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

$\frac{2 x^{3} - 4 x^{2} - 22 x + 44}{x - 2}$

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.

$2$	$2$	$- 4$	$- 22$	$44$
	\xa0			\xa0

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

$2$	$2$	$- 4$	$- 22$	$44$
	\xa0			\xa0
	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(2)$ and place the result of $(4)$ under the next term in the dividend $(- 4)$ .

$2$	$2$	$- 4$	$- 22$	$44$
	\xa0	$4$		\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.

$2$	$2$	$- 4$	$- 22$	$44$
	\xa0	$4$		\xa0
	$2$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(2)$ and place the result of $(0)$ under the next term in the dividend $(- 22)$ .

$2$	$2$	$- 4$	$- 22$	$44$
	\xa0	$4$	$0$	\xa0
	$2$	$0$

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

$2$	$2$	$- 4$	$- 22$	$44$
	\xa0	$4$	$0$	\xa0
	$2$	$0$	$- 22$

Multiply the newest entry in the result $(- 22)$ by the divisor $(2)$ and place the result of $(- 44)$ under the next term in the dividend $(44)$ .

$2$	$2$	$- 4$	$- 22$	$44$
	\xa0	$4$	$0$	$- 44$
	$2$	$0$	$- 22$

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

$2$	$2$	$- 4$	$- 22$	$44$
	\xa0	$4$	$0$	$- 44$
	$2$	$0$	$- 22$	$0$

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

$2 x^{2} + 0 x - 22$

Simplify the quotient polynomial.

$2 x^{2} - 22$

Step 7

Factor $2$ out of $2 x^{2} - 22$ .

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

$(x - 2) (2 (x^{2}) - 22)$

Factor $2$ out of $- 22$ .

$(x - 2) (2 x^{2} + 2 \cdot - 11)$

Factor $2$ out of $2 x^{2} + 2 \cdot - 11$ .

$(x - 2) (2 (x^{2} - 11))$

Step 8

Factor the left side of the equation.

Factor $2$ out of $2 x^{3} - 4 x^{2} - 22 x + 44$ .

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

$2 (x^{3}) - 4 x^{2} - 22 x + 44 = 0$

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

$2 (x^{3}) + 2 (- 2 x^{2}) - 22 x + 44 = 0$

Factor $2$ out of $- 22 x$ .

$2 (x^{3}) + 2 (- 2 x^{2}) + 2 (- 11 x) + 44 = 0$

Factor $2$ out of $44$ .

$2 x^{3} + 2 (- 2 x^{2}) + 2 (- 11 x) + 2 \cdot 22 = 0$

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

$2 (x^{3} - 2 x^{2}) + 2 (- 11 x) + 2 \cdot 22 = 0$

Factor $2$ out of $2 (x^{3} - 2 x^{2}) + 2 (- 11 x)$ .

$2 (x^{3} - 2 x^{2} - 11 x) + 2 \cdot 22 = 0$

Factor $2$ out of $2 (x^{3} - 2 x^{2} - 11 x) + 2 \cdot 22$ .

$2 (x^{3} - 2 x^{2} - 11 x + 22) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((x^{3} - 2 x^{2}) - 11 x + 22) = 0$

Factor out the greatest common factor (GCF) from each group.

$2 (x^{2} (x - 2) - 11 (x - 2)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$2 ((x - 2) (x^{2} - 11)) = 0$

Remove unnecessary parentheses.

$2 (x - 2) (x^{2} - 11) = 0$

Step 9

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

$x - 2 = 0$

$x^{2} - 11 = 0$

Step 10

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 11

Set $x^{2} - 11$ equal to $0$ and solve for $x$ .

Set $x^{2} - 11$ equal to $0$ .

$x^{2} - 11 = 0$

Solve $x^{2} - 11 = 0$ for $x$ .

Add $11$ to both sides of the equation.

$x^{2} = 11$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{11}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{11}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{11}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{11}, - \sqrt{11}$

Step 12

The final solution is all the values that make $2 (x - 2) (x^{2} - 11) = 0$ true.

$x = 2, \sqrt{11}, - \sqrt{11}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = 2, \sqrt{11}, - \sqrt{11}$

Decimal Form:

$x = 2, 3.31662479 \dots, - 3.31662479 \dots$

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=2x^3-4x^2-22x+44

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