Find the Factors Using the Factor Theorem 4x^3-12x^2-x+3 ; x-3

$4 x^{3} - 12 x^{2} - x + 3$ ; $x - 3$

Step 1

Divide $\frac{4 x^{3} - 12 x^{2} - x + 3}{x - 3}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x - 3$ is a factor for $4 x^{3} - 12 x^{2} - x + 3$ . If the remainder is not equal to $0$ , it means that $x - 3$ is not a factor for $4 x^{3} - 12 x^{2} - x + 3$ .

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

$3$	$4$	$- 12$	$- 1$	$3$
	\xa0			\xa0

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

$3$	$4$	$- 12$	$- 1$	$3$
	\xa0			\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(3)$ and place the result of $(12)$ under the next term in the dividend $(- 12)$ .

$3$	$4$	$- 12$	$- 1$	$3$
	\xa0	$12$		\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.

$3$	$4$	$- 12$	$- 1$	$3$
	\xa0	$12$		\xa0
	$4$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(3)$ and place the result of $(0)$ under the next term in the dividend $(- 1)$ .

$3$	$4$	$- 12$	$- 1$	$3$
	\xa0	$12$	$0$	\xa0
	$4$	$0$

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

$3$	$4$	$- 12$	$- 1$	$3$
	\xa0	$12$	$0$	\xa0
	$4$	$0$	$- 1$

Multiply the newest entry in the result $(- 1)$ by the divisor $(3)$ and place the result of $(- 3)$ under the next term in the dividend $(3)$ .

$3$	$4$	$- 12$	$- 1$	$3$
	\xa0	$12$	$0$	$- 3$
	$4$	$0$	$- 1$

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

$3$	$4$	$- 12$	$- 1$	$3$
	\xa0	$12$	$0$	$- 3$
	$4$	$0$	$- 1$	$0$

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

$4 x^{2} + 0 x - 1$

Simplify the quotient polynomial.

$4 x^{2} - 1$

Step 2

The remainder from dividing $\frac{4 x^{3} - 12 x^{2} - x + 3}{x - 3}$ is $0$ , which means that $x - 3$ is a factor for $4 x^{3} - 12 x^{2} - x + 3$ .

$x - 3$ is a factor for $4 x^{3} - 12 x^{2} - x + 3$

Step 3

Find all the possible roots for $4 x^{2} - 1$ .

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.

$p = \pm 1$

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

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}$

Step 4

Set up the next division to determine if $x - \frac{1}{2}$ is a factor of the polynomial $4 x^{2} - 1$ .

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

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x - \frac{1}{2}$ divides evenly into $4 x^{2} - 1$ , $x - \frac{1}{2}$ is a factor of the polynomial and there is a remaining polynomial of $4 x + 2$ .

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

$\frac{1}{2}$	$4$	$0$	$- 1$
	\xa0		\xa0

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

$\frac{1}{2}$	$4$	$0$	$- 1$
	\xa0		\xa0
	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(\frac{1}{2})$ and place the result of $(2)$ under the next term in the dividend $(0)$ .

$\frac{1}{2}$	$4$	$0$	$- 1$
	\xa0	$2$	\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.

$\frac{1}{2}$	$4$	$0$	$- 1$
	\xa0	$2$	\xa0
	$4$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(\frac{1}{2})$ and place the result of $(1)$ under the next term in the dividend $(- 1)$ .

$\frac{1}{2}$	$4$	$0$	$- 1$
	\xa0	$2$	$1$
	$4$	$2$

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

$\frac{1}{2}$	$4$	$0$	$- 1$
	\xa0	$2$	$1$
	$4$	$2$	$0$

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

$(4) x + 2$

Simplify the quotient polynomial.

$4 x + 2$

Step 6

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

$4 x + 2$

Step 7

The factored polynomial is $(x - 3) (x - \frac{1}{2}) (4 x + 2)$ .

$(x - 3) (x - \frac{1}{2}) (4 x + 2)$

Find the Factors Using the Factor Theorem 4x^3-12x^2-x+3 ; x-3

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