Find the Roots/Zeros Using the Rational Roots Test 2x^4-11x^3+26x^2-66x+84=0

$2 x^{4} - 11 x^{3} + 26 x^{2} - 66 x + 84 = 0$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 7, \pm 12, \pm 14, \pm 21, \pm 28, \pm 42, \pm 84$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm 4, \pm 6, \pm 7, \pm \frac{7}{2}, \pm 12, \pm 14, \pm 21, \pm \frac{21}{2}, \pm 28, \pm 42, \pm 84$

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.

$2 {(2)}^{4} - 11 {(2)}^{3} + 26 {(2)}^{2} - 66 \cdot 2 + 84$

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 ${(2)}^{4}$ by adding the exponents.

Multiply $2$ by ${(2)}^{4}$ .

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

$2^{1} {(2)}^{4} - 11 {(2)}^{3} + 26 {(2)}^{2} - 66 \cdot 2 + 84$

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

$2^{1 + 4} - 11 {(2)}^{3} + 26 {(2)}^{2} - 66 \cdot 2 + 84$

Add $1$ and $4$ .

$2^{5} - 11 {(2)}^{3} + 26 {(2)}^{2} - 66 \cdot 2 + 84$

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

$32 - 11 {(2)}^{3} + 26 {(2)}^{2} - 66 \cdot 2 + 84$

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

$32 - 11 \cdot 8 + 26 {(2)}^{2} - 66 \cdot 2 + 84$

Multiply $- 11$ by $8$ .

$32 - 88 + 26 {(2)}^{2} - 66 \cdot 2 + 84$

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

$32 - 88 + 26 \cdot 4 - 66 \cdot 2 + 84$

Multiply $26$ by $4$ .

$32 - 88 + 104 - 66 \cdot 2 + 84$

Multiply $- 66$ by $2$ .

$32 - 88 + 104 - 132 + 84$

Simplify by adding and subtracting.

Subtract $88$ from $32$ .

$- 56 + 104 - 132 + 84$

Add $- 56$ and $104$ .

$48 - 132 + 84$

Subtract $132$ from $48$ .

$- 84 + 84$

Add $- 84$ and $84$ .

$0$

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.

$\frac{2 x^{4} - 11 x^{3} + 26 x^{2} - 66 x + 84}{x - 2}$

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.

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\xa0				\xa0

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

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\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 $(- 11)$ .

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\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.

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\xa0	$4$			\xa0
	$2$	$- 7$

Multiply the newest entry in the result $(- 7)$ by the divisor $(2)$ and place the result of $(- 14)$ under the next term in the dividend $(26)$ .

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\xa0	$4$	$- 14$		\xa0
	$2$	$- 7$

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

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\xa0	$4$	$- 14$		\xa0
	$2$	$- 7$	$12$

Multiply the newest entry in the result $(12)$ by the divisor $(2)$ and place the result of $(24)$ under the next term in the dividend $(- 66)$ .

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\xa0	$4$	$- 14$	$24$	\xa0
	$2$	$- 7$	$12$

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

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\xa0	$4$	$- 14$	$24$	\xa0
	$2$	$- 7$	$12$	$- 42$

Multiply the newest entry in the result $(- 42)$ by the divisor $(2)$ and place the result of $(- 84)$ under the next term in the dividend $(84)$ .

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\xa0	$4$	$- 14$	$24$	$- 84$
	$2$	$- 7$	$12$	$- 42$

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

$2$	$2$	$- 11$	$26$	$- 66$	$84$
	\xa0	$4$	$- 14$	$24$	$- 84$
	$2$	$- 7$	$12$	$- 42$	$0$

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

$2 x^{3} + - 7 x^{2} + (12) x - 42$

Simplify the quotient polynomial.

$2 x^{3} - 7 x^{2} + 12 x - 42$

Step 7

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 2) (2 x^{3} - 7 x^{2}) + 12 x - 42$

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

$(x - 2) + x^{2} (2 x - 7) + 6 (2 x - 7)$

$(x - 2) x^{2} (2 x - 7) + 6 (2 x - 7)$

Step 8

Factor the polynomial by factoring out the greatest common factor, $2 x - 7$ .

$(x - 2) (2 x - 7) (x^{2} + 6)$

Step 9

Factor the left side of the equation.

Regroup terms.

$- 11 x^{3} - 66 x + 2 x^{4} + 26 x^{2} + 84 = 0$

Factor $- 11 x$ out of $- 11 x^{3} - 66 x$ .

Factor $- 11 x$ out of $- 11 x^{3}$ .

$- 11 x \cdot x^{2} - 66 x + 2 x^{4} + 26 x^{2} + 84 = 0$

Factor $- 11 x$ out of $- 66 x$ .

$- 11 x \cdot x^{2} - 11 x \cdot 6 + 2 x^{4} + 26 x^{2} + 84 = 0$

Factor $- 11 x$ out of $- 11 x (x^{2}) - 11 x (6)$ .

$- 11 x (x^{2} + 6) + 2 x^{4} + 26 x^{2} + 84 = 0$

Factor $2$ out of $2 x^{4} + 26 x^{2} + 84$ .

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

$- 11 x (x^{2} + 6) + 2 (x^{4}) + 26 x^{2} + 84 = 0$

Factor $2$ out of $26 x^{2}$ .

$- 11 x (x^{2} + 6) + 2 (x^{4}) + 2 (13 x^{2}) + 84 = 0$

Factor $2$ out of $84$ .

$- 11 x (x^{2} + 6) + 2 x^{4} + 2 (13 x^{2}) + 2 \cdot 42 = 0$

Factor $2$ out of $2 x^{4} + 2 (13 x^{2})$ .

$- 11 x (x^{2} + 6) + 2 (x^{4} + 13 x^{2}) + 2 \cdot 42 = 0$

Factor $2$ out of $2 (x^{4} + 13 x^{2}) + 2 \cdot 42$ .

$- 11 x (x^{2} + 6) + 2 (x^{4} + 13 x^{2} + 42) = 0$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- 11 x (x^{2} + 6) + 2 ({(x^{2})}^{2} + 13 x^{2} + 42) = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 11 x (x^{2} + 6) + 2 (u^{2} + 13 u + 42) = 0$

Factor $u^{2} + 13 u + 42$ 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 $42$ and whose sum is $13$ .

$6, 7$

Write the factored form using these integers.

$- 11 x (x^{2} + 6) + 2 ((u + 6) (u + 7)) = 0$

Factor.

Replace all occurrences of $u$ with $x^{2}$ .

$- 11 x (x^{2} + 6) + 2 ((x^{2} + 6) (x^{2} + 7)) = 0$

Remove unnecessary parentheses.

$- 11 x (x^{2} + 6) + 2 (x^{2} + 6) (x^{2} + 7) = 0$

Factor $x^{2} + 6$ out of $- 11 x (x^{2} + 6) + 2 (x^{2} + 6) (x^{2} + 7)$ .

Factor $x^{2} + 6$ out of $- 11 x (x^{2} + 6)$ .

$(x^{2} + 6) (- 11 x) + 2 (x^{2} + 6) (x^{2} + 7) = 0$

Factor $x^{2} + 6$ out of $2 (x^{2} + 6) (x^{2} + 7)$ .

$(x^{2} + 6) (- 11 x) + (x^{2} + 6) (2 (x^{2} + 7)) = 0$

Factor $x^{2} + 6$ out of $(x^{2} + 6) (- 11 x) + (x^{2} + 6) (2 (x^{2} + 7))$ .

$(x^{2} + 6) (- 11 x + 2 (x^{2} + 7)) = 0$

Apply the distributive property.

$(x^{2} + 6) (- 11 x + 2 x^{2} + 2 \cdot 7) = 0$

Multiply $2$ by $7$ .

$(x^{2} + 6) (- 11 x + 2 x^{2} + 14) = 0$

Reorder terms.

$(x^{2} + 6) (2 x^{2} - 11 x + 14) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 14 = 28$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 x$ .

$(x^{2} + 6) (2 x^{2} - 11 x + 14) = 0$

Rewrite $- 11$ as $- 4$ plus $- 7$

$(x^{2} + 6) (2 x^{2} + (- 4 - 7) x + 14) = 0$

Apply the distributive property.

$(x^{2} + 6) (2 x^{2} - 4 x - 7 x + 14) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{2} + 6) ((2 x^{2} - 4 x) - 7 x + 14) = 0$

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

$(x^{2} + 6) (2 x (x - 2) - 7 (x - 2)) = 0$

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

$(x^{2} + 6) ((x - 2) (2 x - 7)) = 0$

Remove unnecessary parentheses.

$(x^{2} + 6) (x - 2) (2 x - 7) = 0$

Step 10

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

$x^{2} + 6 = 0$

$x - 2 = 0$

$2 x - 7 = 0$

Step 11

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

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

$x^{2} + 6 = 0$

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

Subtract $6$ from both sides of the equation.

$x^{2} = - 6$

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

$x = \pm \sqrt{- 6}$

Simplify $\pm \sqrt{- 6}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$x = \pm \sqrt{- 1 (6)}$

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

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

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

$x = \pm i \sqrt{6}$

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.

$x = i \sqrt{6}$

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

$x = - i \sqrt{6}$

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

$x = i \sqrt{6}, - i \sqrt{6}$

Step 12

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 13

Set $2 x - 7$ equal to $0$ and solve for $x$ .

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Solve $2 x - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$2 x = 7$

Divide each term in $2 x = 7$ by $2$ and simplify.

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 14

The final solution is all the values that make $(x^{2} + 6) (x - 2) (2 x - 7) = 0$ true.

$x = i \sqrt{6}, - i \sqrt{6}, 2, \frac{7}{2}$

Step 15

Find the Roots/Zeros Using the Rational Roots Test 2x^4-11x^3+26x^2-66x+84=0

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