Precalculus Examples

$2 x^{2} + x - 3$ , $x - 1$

Step 1

Divide $\frac{2 x^{2} + x - 3}{x - 1}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x - 1$ is a factor for $2 x^{2} + x - 3$ . If the remainder is not equal to $0$ , it means that $x - 1$ is not a factor for $2 x^{2} + x - 3$ .

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Step 1.1

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

$1$	$2$	$1$	$- 3$

Step 1.2

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

$1$	$2$	$1$	$- 3$

	$2$

Step 1.3

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

$1$	$2$	$1$	$- 3$
		$2$
	$2$

Step 1.4

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

$1$	$2$	$1$	$- 3$
		$2$
	$2$	$3$

Step 1.5

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

$1$	$2$	$1$	$- 3$
		$2$	$3$
	$2$	$3$

Step 1.6

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

$1$	$2$	$1$	$- 3$
		$2$	$3$
	$2$	$3$	$0$

Step 1.7

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

$(2) x + 3$

Step 1.8

Simplify the quotient polynomial.

$2 x + 3$

Step 2

The remainder from dividing $\frac{2 x^{2} + x - 3}{x - 1}$ is $0$ , which means that $x - 1$ is a factor for $2 x^{2} + x - 3$ .

$x - 1$ is a factor for $2 x^{2} + x - 3$

Step 3

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

$2 x + 3$

Step 4

The factored polynomial is $(x - 1) (2 x + 3)$ .

$(x - 1) (2 x + 3)$

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