Factor 4x^3-8x^2+x+3

$4 x^{3} - 8 x^{2} + x + 3$

Step 1

Factor $4 x^{3} - 8 x^{2} + x + 3$ using the rational roots test.

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.

$p = \pm 1, \pm 3 q = \pm 1, \pm 4, \pm 2$

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 3, \pm 0.75, \pm 1.5$

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

Substitute $- 0.5$ into the polynomial.

$4 {(- 0.5)}^{3} - 8 {(- 0.5)}^{2} - 0.5 + 3$

Raise $- 0.5$ to the power of $3$ .

$4 \cdot - 0.125 - 8 {(- 0.5)}^{2} - 0.5 + 3$

Multiply $4$ by $- 0.125$ .

$- 0.5 - 8 {(- 0.5)}^{2} - 0.5 + 3$

Raise $- 0.5$ to the power of $2$ .

$- 0.5 - 8 \cdot 0.25 - 0.5 + 3$

Multiply $- 8$ by $0.25$ .

$- 0.5 - 2 - 0.5 + 3$

Subtract $2$ from $- 0.5$ .

$- 2.5 - 0.5 + 3$

Subtract $0.5$ from $- 2.5$ .

$- 3 + 3$

Add $- 3$ and $3$ .

$0$

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 8 x^{2} + x + 3}{2 x + 1}$

Divide $4 x^{3} - 8 x^{2} + x + 3$ by $2 x + 1$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$2 x$

$1$

$4 x^{3}$

$8 x^{2}$

$x$

$3$

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			+	$4 x^{3}$	+	$2 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + 2 x^{2}$

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$
					-	$10 x^{2}$	-	$5 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 5 x$

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$
					+	$10 x^{2}$	+	$5 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$
					+	$10 x^{2}$	+	$5 x$
							+	$6 x$

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$	-	$5 x$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$
					+	$10 x^{2}$	+	$5 x$
							+	$6 x$	+	$3$

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	-	$5 x$	+	$3$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$
					+	$10 x^{2}$	+	$5 x$
							+	$6 x$	+	$3$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$	+	$3$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$
					+	$10 x^{2}$	+	$5 x$
							+	$6 x$	+	$3$
							+	$6 x$	+	$3$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + 3$

				$2 x^{2}$	-	$5 x$	+	$3$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$
					+	$10 x^{2}$	+	$5 x$
							+	$6 x$	+	$3$
							-	$6 x$	-	$3$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$5 x$	+	$3$
$2 x$	+	$1$		$4 x^{3}$	-	$8 x^{2}$	+	$x$	+	$3$
			-	$4 x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	+	$x$
					+	$10 x^{2}$	+	$5 x$
							+	$6 x$	+	$3$
							-	$6 x$	-	$3$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$2 x^{2} - 5 x + 3$

Write $4 x^{3} - 8 x^{2} + x + 3$ as a set of factors.

$(2 x + 1) (2 x^{2} - 5 x + 3)$

Step 2

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$(2 x + 1) (2 x^{2} - 5 (x) + 3)$

Rewrite $- 5$ as $- 2$ plus $- 3$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Remove unnecessary parentheses.

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

Factor 4x^3-8x^2+x+3

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