Find the Factors Using the Factor Theorem 2x^3+9x^2-4x-18 ; 2x+9

$2 x^{3} + 9 x^{2} - 4 x - 18$ ; $2 x + 9$

Step 1

Divide $\frac{2 x^{3} + 9 x^{2} - 4 x - 18}{2 x + 9}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $2 x + 9$ is a factor for $2 x^{3} + 9 x^{2} - 4 x - 18$ . If the remainder is not equal to $0$ , it means that $2 x + 9$ is not a factor for $2 x^{3} + 9 x^{2} - 4 x - 18$ .

Divide each term in the denominator by $2$ to make the coefficient of linear factor variable $1$ .

$\frac{1}{2} \cdot \frac{2 x^{3} + 9 x^{2} - 4 x - 18}{\frac{2 x + 9}{2}}$

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- \frac{9}{2}$	$2$	$9$	$- 4$	$- 18$
	\xa0			\xa0

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

$- \frac{9}{2}$	$2$	$9$	$- 4$	$- 18$
	\xa0			\xa0
	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{9}{2})$ and place the result of $(- 9)$ under the next term in the dividend $(9)$ .

$- \frac{9}{2}$	$2$	$9$	$- 4$	$- 18$
	\xa0	$- 9$		\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.

$- \frac{9}{2}$	$2$	$9$	$- 4$	$- 18$
	\xa0	$- 9$		\xa0
	$2$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- \frac{9}{2})$ and place the result of $(0)$ under the next term in the dividend $(- 4)$ .

$- \frac{9}{2}$	$2$	$9$	$- 4$	$- 18$
	\xa0	$- 9$	$0$	\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.

$- \frac{9}{2}$	$2$	$9$	$- 4$	$- 18$
	\xa0	$- 9$	$0$	\xa0
	$2$	$0$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(- \frac{9}{2})$ and place the result of $(18)$ under the next term in the dividend $(- 18)$ .

$- \frac{9}{2}$	$2$	$9$	$- 4$	$- 18$
	\xa0	$- 9$	$0$	$18$
	$2$	$0$	$- 4$

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

$- \frac{9}{2}$	$2$	$9$	$- 4$	$- 18$
	\xa0	$- 9$	$0$	$18$
	$2$	$0$	$- 4$	$0$

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

$(\frac{1}{2}) \cdot (2 x^{2} + 0 x - 4)$

Simplify the quotient polynomial.

$(\frac{1}{2}) \cdot (2 x^{2} - 4)$

Simplify.

Apply the distributive property.

$\frac{1}{2} \cdot (2 x^{2}) + \frac{1}{2} \cdot - 4$

Cancel the common factor of $2$ .

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

$\frac{1}{2} \cdot (2 (x^{2})) + \frac{1}{2} \cdot - 4$

Cancel the common factor.

$\frac{1}{2} \cdot (2 x^{2}) + \frac{1}{2} \cdot - 4$

Rewrite the expression.

$x^{2} + \frac{1}{2} \cdot - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $- 4$ .

$x^{2} + \frac{1}{2} \cdot (2 (- 2))$

Cancel the common factor.

$x^{2} + \frac{1}{2} \cdot (2 \cdot - 2)$

Rewrite the expression.

$x^{2} - 2$

Step 2

The remainder from dividing $\frac{2 x^{3} + 9 x^{2} - 4 x - 18}{2 x + 9}$ is $0$ , which means that $2 x + 9$ is a factor for $2 x^{3} + 9 x^{2} - 4 x - 18$ .

$2 x + 9$ is a factor for $2 x^{3} + 9 x^{2} - 4 x - 18$

Step 3

Find all the possible roots for $x^{2} - 2$ .

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.

$p = \pm 1, \pm 2$

$q = \pm 1$

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

$\pm 1, \pm 2$

Step 4

The final factor is the only factor left over from the synthetic division.

$x^{2} - 2$

Step 5

The factored polynomial is $(2 x + 9) (x^{2} - 2)$ .

$(2 x + 9) (x^{2} - 2)$

Find the Factors Using the Factor Theorem 2x^3+9x^2-4x-18 ; 2x+9

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