Factor 22y^4-33y^2+11y

$22 y^{4} - 33 y^{2} + 11 y$

Step 1

Factor $11 y$ out of $22 y^{4} - 33 y^{2} + 11 y$ .

Factor $11 y$ out of $22 y^{4}$ .

$11 y (2 y^{3}) - 33 y^{2} + 11 y$

Factor $11 y$ out of $- 33 y^{2}$ .

$11 y (2 y^{3}) + 11 y (- 3 y) + 11 y$

Factor $11 y$ out of $11 y$ .

$11 y (2 y^{3}) + 11 y (- 3 y) + 11 y (1)$

Factor $11 y$ out of $11 y (2 y^{3}) + 11 y (- 3 y)$ .

$11 y (2 y^{3} - 3 y) + 11 y (1)$

Factor $11 y$ out of $11 y (2 y^{3} - 3 y) + 11 y (1)$ .

$11 y (2 y^{3} - 3 y + 1)$

Step 2

Factor.

Factor $2 y^{3} - 3 y + 1$ 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.

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

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

$\pm 1, \pm 0.5$

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

Substitute $1$ into the polynomial.

$2 \cdot 1^{3} - 3 \cdot 1 + 1$

Raise $1$ to the power of $3$ .

$2 \cdot 1 - 3 \cdot 1 + 1$

Multiply $2$ by $1$ .

$2 - 3 \cdot 1 + 1$

Multiply $- 3$ by $1$ .

$2 - 3 + 1$

Subtract $3$ from $2$ .

$- 1 + 1$

Add $- 1$ and $1$ .

$0$

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

$\frac{2 y^{3} - 3 y + 1}{y - 1}$

Divide $2 y^{3} - 3 y + 1$ by $y - 1$ .

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

$y$

$1$

$2 y^{3}$

$0 y^{2}$

$3 y$

$1$

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

					$2 y^{2}$
	$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$

Multiply the new quotient term by the divisor.

				$2 y^{2}$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			+	$2 y^{3}$	-	$2 y^{2}$

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

				$2 y^{2}$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$

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

				$2 y^{2}$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$

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

				$2 y^{2}$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$

Divide the highest order term in the dividend $2 y^{2}$ by the highest order term in divisor $y$ .

				$2 y^{2}$	+	$2 y$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$

Multiply the new quotient term by the divisor.

				$2 y^{2}$	+	$2 y$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$
					+	$2 y^{2}$	-	$2 y$

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

				$2 y^{2}$	+	$2 y$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$
					-	$2 y^{2}$	+	$2 y$

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

				$2 y^{2}$	+	$2 y$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$
					-	$2 y^{2}$	+	$2 y$
							-	$y$

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

				$2 y^{2}$	+	$2 y$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$
					-	$2 y^{2}$	+	$2 y$
							-	$y$	+	$1$

Divide the highest order term in the dividend $- y$ by the highest order term in divisor $y$ .

				$2 y^{2}$	+	$2 y$	-	$1$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$
					-	$2 y^{2}$	+	$2 y$
							-	$y$	+	$1$

Multiply the new quotient term by the divisor.

				$2 y^{2}$	+	$2 y$	-	$1$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$
					-	$2 y^{2}$	+	$2 y$
							-	$y$	+	$1$
							-	$y$	+	$1$

The expression needs to be subtracted from the dividend, so change all the signs in $- y + 1$

				$2 y^{2}$	+	$2 y$	-	$1$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$
					-	$2 y^{2}$	+	$2 y$
							-	$y$	+	$1$
							+	$y$	-	$1$

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

				$2 y^{2}$	+	$2 y$	-	$1$
$y$	-	$1$		$2 y^{3}$	+	$0 y^{2}$	-	$3 y$	+	$1$
			-	$2 y^{3}$	+	$2 y^{2}$
					+	$2 y^{2}$	-	$3 y$
					-	$2 y^{2}$	+	$2 y$
							-	$y$	+	$1$
							+	$y$	-	$1$
										$0$

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

$2 y^{2} + 2 y - 1$

Write $2 y^{3} - 3 y + 1$ as a set of factors.

$11 y ((y - 1) (2 y^{2} + 2 y - 1))$

Remove unnecessary parentheses.

$11 y (y - 1) (2 y^{2} + 2 y - 1)$

Factor 22y^4-33y^2+11y

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