Find the Roots/Zeros Using the Rational Roots Test 4x^5+16x^4-385x^3-4x^2+96x

$4 x^{5} + 16 x^{4} - 385 x^{3} - 4 x^{2} + 96 x$

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.

$p = \pm 0$

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

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

$0$

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.

$4 {(0)}^{5} + 16 {(0)}^{4} - 385 {(0)}^{3} - 4 {(0)}^{2} + 96 (0)$

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

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$4 \cdot 0 + 16 {(0)}^{4} - 385 {(0)}^{3} - 4 {(0)}^{2} + 96 (0)$

Multiply $4$ by $0$ .

$0 + 16 {(0)}^{4} - 385 {(0)}^{3} - 4 {(0)}^{2} + 96 (0)$

Raising $0$ to any positive power yields $0$ .

$0 + 16 \cdot 0 - 385 {(0)}^{3} - 4 {(0)}^{2} + 96 (0)$

Multiply $16$ by $0$ .

$0 + 0 - 385 {(0)}^{3} - 4 {(0)}^{2} + 96 (0)$

Raising $0$ to any positive power yields $0$ .

$0 + 0 - 385 \cdot 0 - 4 {(0)}^{2} + 96 (0)$

Multiply $- 385$ by $0$ .

$0 + 0 + 0 - 4 {(0)}^{2} + 96 (0)$

Raising $0$ to any positive power yields $0$ .

$0 + 0 + 0 - 4 \cdot 0 + 96 (0)$

Multiply $- 4$ by $0$ .

$0 + 0 + 0 + 0 + 96 (0)$

Multiply $96$ by $0$ .

$0 + 0 + 0 + 0 + 0$

$0 + 0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$0 + 0 + 0 + 0$

Add $0$ and $0$ .

$0 + 0 + 0$

Add $0$ and $0$ .

$0 + 0$

Add $0$ and $0$ .

$0$

$0$

$0$

Since $0$ is a known root, divide the polynomial by $x + 0$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{5} + 16 x^{4} - 385 x^{3} - 4 x^{2} + 96 x}{x + 0}$

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.

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0					\xa0

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

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0					\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(16)$ .

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$				\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.

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$				\xa0
	$4$	$16$

Multiply the newest entry in the result $(16)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 385)$ .

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$	$0$			\xa0
	$4$	$16$

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

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$	$0$			\xa0
	$4$	$16$	$- 385$

Multiply the newest entry in the result $(- 385)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 4)$ .

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$	$0$	$0$		\xa0
	$4$	$16$	$- 385$

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

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$	$0$	$0$		\xa0
	$4$	$16$	$- 385$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(96)$ .

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$	$0$	$0$	$0$	\xa0
	$4$	$16$	$- 385$	$- 4$

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

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$	$0$	$0$	$0$	\xa0
	$4$	$16$	$- 385$	$- 4$	$96$

Multiply the newest entry in the result $(96)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$	$0$	$0$	$0$	$0$
	$4$	$16$	$- 385$	$- 4$	$96$

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

$0$	$4$	$16$	$- 385$	$- 4$	$96$	$0$
	\xa0	$0$	$0$	$0$	$0$	$0$
	$4$	$16$	$- 385$	$- 4$	$96$	$0$

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

$4 x^{4} + 16 x^{3} - 385 x^{2} + - 4 x + 96$

Simplify the quotient polynomial.

$4 x^{4} + 16 x^{3} - 385 x^{2} - 4 x + 96$

$4 x^{4} + 16 x^{3} - 385 x^{2} - 4 x + 96$

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Regroup terms.

$16 x^{3} - 4 x + 4 x^{4} - 385 x^{2} + 96 = 0$

Factor $4 x$ out of $16 x^{3} - 4 x$ .

Factor $4 x$ out of $16 x^{3}$ .

$4 x (4 x^{2}) - 4 x + 4 x^{4} - 385 x^{2} + 96 = 0$

Factor $4 x$ out of $- 4 x$ .

$4 x (4 x^{2}) + 4 x (- 1) + 4 x^{4} - 385 x^{2} + 96 = 0$

Factor $4 x$ out of $4 x (4 x^{2}) + 4 x (- 1)$ .

$4 x (4 x^{2} - 1) + 4 x^{4} - 385 x^{2} + 96 = 0$

$4 x (4 x^{2} - 1) + 4 x^{4} - 385 x^{2} + 96 = 0$

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

$4 x ({(2 x)}^{2} - 1) + 4 x^{4} - 385 x^{2} + 96 = 0$

Rewrite $1$ as $1^{2}$ .

$4 x ({(2 x)}^{2} - 1^{2}) + 4 x^{4} - 385 x^{2} + 96 = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

$4 x ((2 x + 1) (2 x - 1)) + 4 x^{4} - 385 x^{2} + 96 = 0$

Remove unnecessary parentheses.

$4 x (2 x + 1) (2 x - 1) + 4 x^{4} - 385 x^{2} + 96 = 0$

$4 x (2 x + 1) (2 x - 1) + 4 x^{4} - 385 x^{2} + 96 = 0$

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

$4 x (2 x + 1) (2 x - 1) + 4 {(x^{2})}^{2} - 385 x^{2} + 96 = 0$

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

$4 x (2 x + 1) (2 x - 1) + 4 u^{2} - 385 u + 96 = 0$

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 $a \cdot c = 4 \cdot 96 = 384$ and whose sum is $b = - 385$ .

Factor $- 385$ out of $- 385 u$ .

$4 x (2 x + 1) (2 x - 1) + 4 u^{2} - 385 u + 96 = 0$

Rewrite $- 385$ as $- 1$ plus $- 384$

$4 x (2 x + 1) (2 x - 1) + 4 u^{2} + (- 1 - 384) u + 96 = 0$

Apply the distributive property.

$4 x (2 x + 1) (2 x - 1) + 4 u^{2} - 1 u - 384 u + 96 = 0$

$4 x (2 x + 1) (2 x - 1) + 4 u^{2} - 1 u - 384 u + 96 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 x (2 x + 1) (2 x - 1) + 4 u^{2} - 1 u - 384 u + 96 = 0$

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

$4 x (2 x + 1) (2 x - 1) + u (4 u - 1) - 96 (4 u - 1) = 0$

$4 x (2 x + 1) (2 x - 1) + u (4 u - 1) - 96 (4 u - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$4 x (2 x + 1) (2 x - 1) + (4 u - 1) (u - 96) = 0$

$4 x (2 x + 1) (2 x - 1) + (4 u - 1) (u - 96) = 0$

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

$4 x (2 x + 1) (2 x - 1) + (4 x^{2} - 1) (x^{2} - 96) = 0$

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

$4 x (2 x + 1) (2 x - 1) + ({(2 x)}^{2} - 1) (x^{2} - 96) = 0$

Rewrite $1$ as $1^{2}$ .

$4 x (2 x + 1) (2 x - 1) + ({(2 x)}^{2} - 1^{2}) (x^{2} - 96) = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

$4 x (2 x + 1) (2 x - 1) + (2 x + 1) (2 x - 1) (x^{2} - 96) = 0$

Factor $(2 x + 1) (2 x - 1)$ out of $4 x (2 x + 1) (2 x - 1) + (2 x + 1) (2 x - 1) (x^{2} - 96)$ .

Factor $(2 x + 1) (2 x - 1)$ out of $4 x (2 x + 1) (2 x - 1)$ .

$(2 x + 1) (2 x - 1) (4 x) + (2 x + 1) (2 x - 1) (x^{2} - 96) = 0$

Factor $(2 x + 1) (2 x - 1)$ out of $(2 x + 1) (2 x - 1) (4 x) + (2 x + 1) (2 x - 1) (x^{2} - 96)$ .

$(2 x + 1) (2 x - 1) (4 x + x^{2} - 96) = 0$

$(2 x + 1) (2 x - 1) (4 x + x^{2} - 96) = 0$

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(2 x + 1) (2 x - 1) (4 u + u^{2} - 96) = 0$

Factor $4 u + u^{2} - 96$ 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 $- 96$ and whose sum is $4$ .

$- 8, 12$

Write the factored form using these integers.

$(2 x + 1) (2 x - 1) ((u - 8) (u + 12)) = 0$

$(2 x + 1) (2 x - 1) ((u - 8) (u + 12)) = 0$

Factor.

Replace all occurrences of $u$ with $x$ .

$(2 x + 1) (2 x - 1) ((x - 8) (x + 12)) = 0$

Remove unnecessary parentheses.

$(2 x + 1) (2 x - 1) (x - 8) (x + 12) = 0$

$(2 x + 1) (2 x - 1) (x - 8) (x + 12) = 0$

$(2 x + 1) (2 x - 1) (x - 8) (x + 12) = 0$

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

$2 x + 1 = 0$

$2 x - 1 = 0$

$x - 8 = 0$

$x + 12 = 0$

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 $x$ .

Subtract $1$ from 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 $1$ .

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

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

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

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

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

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 $1$ .

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

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

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

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

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

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

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

$x = 8$

Set $x + 12$ equal to $0$ and solve for $x$ .

Set $x + 12$ equal to $0$ .

$x + 12 = 0$

Subtract $12$ from both sides of the equation.

$x = - 12$

$x = - 12$

The final solution is all the values that make $(2 x + 1) (2 x - 1) (x - 8) (x + 12) = 0$ true.

$x = - \frac{1}{2}, \frac{1}{2}, 8, - 12$

$x = - \frac{1}{2}, \frac{1}{2}, 8, - 12$

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

$(x + 0) (x + \frac{1}{2}) (x - \frac{1}{2}) (x - 8) (x + 12)$

These are the roots (zeros) of the polynomial $4 x^{5} + 16 x^{4} - 385 x^{3} - 4 x^{2} + 96 x$ .

$x = 0, - \frac{1}{2}, \frac{1}{2}, 8, - 12$

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