Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4+2x^3-3x^2-6x

$f (x) = x^{4} + 2 x^{3} - 3 x^{2} - 6 x$

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.

$p = \pm 0$

$q = \pm 1$

Step 2

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

$0$

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.

${(0)}^{4} + 2 {(0)}^{3} - 3 {(0)}^{2} - 6 \cdot 0$

Step 4

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 $0$ .

$0 + 2 {(0)}^{3} - 3 {(0)}^{2} - 6 \cdot 0$

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

$0 + 2 \cdot 0 - 3 {(0)}^{2} - 6 \cdot 0$

Multiply $2$ by $0$ .

$0 + 0 - 3 {(0)}^{2} - 6 \cdot 0$

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

$0 + 0 - 3 \cdot 0 - 6 \cdot 0$

Multiply $- 3$ by $0$ .

$0 + 0 + 0 - 6 \cdot 0$

Multiply $- 6$ by $0$ .

$0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$0 + 0 + 0$

Add $0$ and $0$ .

$0 + 0$

Add $0$ and $0$ .

$0$

Step 5

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.

$\frac{x^{4} + 2 x^{3} - 3 x^{2} - 6 x}{x + 0}$

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.

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0				\xa0

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

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(2)$ .

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0	$0$			\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.

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0	$0$			\xa0
	$1$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 3)$ .

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0	$0$	$0$		\xa0
	$1$	$2$

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

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0	$0$	$0$		\xa0
	$1$	$2$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 6)$ .

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$1$	$2$	$- 3$

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

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$1$	$2$	$- 3$	$- 6$

Multiply the newest entry in the result $(- 6)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$1$	$2$	$- 3$	$- 6$

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

$0$	$1$	$2$	$- 3$	$- 6$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$1$	$2$	$- 3$	$- 6$	$0$

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

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

Simplify the quotient polynomial.

$x^{3} + 2 x^{2} - 3 x - 6$

Step 7

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

Step 8

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

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

Step 9

Factor the left side of the equation.

Factor $x$ out of $x^{4} + 2 x^{3} - 3 x^{2} - 6 x$ .

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

$x \cdot x^{3} + 2 x^{3} - 3 x^{2} - 6 x = 0$

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

$x \cdot x^{3} + x (2 x^{2}) - 3 x^{2} - 6 x = 0$

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

$x \cdot x^{3} + x (2 x^{2}) + x (- 3 x) - 6 x = 0$

Factor $x$ out of $- 6 x$ .

$x \cdot x^{3} + x (2 x^{2}) + x (- 3 x) + x \cdot - 6 = 0$

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

$x (x^{3} + 2 x^{2}) + x (- 3 x) + x \cdot - 6 = 0$

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

$x (x^{3} + 2 x^{2} - 3 x) + x \cdot - 6 = 0$

Factor $x$ out of $x (x^{3} + 2 x^{2} - 3 x) + x \cdot - 6$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor.

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

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

Remove unnecessary parentheses.

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

Step 10

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

$x = 0$

$x + 2 = 0$

$x^{2} - 3 = 0$

Step 11

Set $x$ equal to $0$ .

$x = 0$

Step 12

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 13

Set $x^{2} - 3$ equal to $0$ and solve for $x$ .

Set $x^{2} - 3$ equal to $0$ .

$x^{2} - 3 = 0$

Solve $x^{2} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$x^{2} = 3$

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

$x = \pm \sqrt{3}$

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.

$x = \sqrt{3}$

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

$x = - \sqrt{3}$

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

$x = \sqrt{3}, - \sqrt{3}$

Step 14

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

$x = 0, - 2, \sqrt{3}, - \sqrt{3}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$x = 0, - 2, \sqrt{3}, - \sqrt{3}$

Decimal Form:

$x = 0, - 2, 1.73205080 \dots, - 1.73205080 \dots$

Step 16

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4+2x^3-3x^2-6x

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