Simplify (9y^4+14y^2-8)/(3y+2)

$\frac{9 y^{4} + 14 y^{2} - 8}{3 y + 2}$

Step 1

Simplify the numerator.

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

$\frac{9 {(y^{2})}^{2} + 14 y^{2} - 8}{3 y + 2}$

Let $u = y^{2}$ . Substitute $u$ for all occurrences of $y^{2}$ .

$\frac{9 u^{2} + 14 u - 8}{3 y + 2}$

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 $a \cdot c = 9 \cdot - 8 = - 72$ and whose sum is $b = 14$ .

Factor $14$ out of $14 u$ .

$\frac{9 u^{2} + 14 (u) - 8}{3 y + 2}$

Rewrite $14$ as $- 4$ plus $18$

$\frac{9 u^{2} + (- 4 + 18) u - 8}{3 y + 2}$

Apply the distributive property.

$\frac{9 u^{2} - 4 u + 18 u - 8}{3 y + 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(9 u^{2} - 4 u) + 18 u - 8}{3 y + 2}$

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

$\frac{u (9 u - 4) + 2 (9 u - 4)}{3 y + 2}$

Factor the polynomial by factoring out the greatest common factor, $9 u - 4$ .

$\frac{(9 u - 4) (u + 2)}{3 y + 2}$

Replace all occurrences of $u$ with $y^{2}$ .

$\frac{(9 y^{2} - 4) (y^{2} + 2)}{3 y + 2}$

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

$\frac{({(3 y)}^{2} - 4) (y^{2} + 2)}{3 y + 2}$

Rewrite $4$ as $2^{2}$ .

$\frac{({(3 y)}^{2} - 2^{2}) (y^{2} + 2)}{3 y + 2}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 y$ and $b = 2$ .

$\frac{(3 y + 2) (3 y - 2) (y^{2} + 2)}{3 y + 2}$

Step 2

Cancel the common factor of $3 y + 2$ .

Cancel the common factor.

$\frac{(3 y + 2) (3 y - 2) (y^{2} + 2)}{3 y + 2}$

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

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

Step 3

Expand $(3 y - 2) (y^{2} + 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$3 y \cdot y^{2} + 3 y \cdot 2 - 2 (y^{2} + 2)$

Apply the distributive property.

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

Step 4

Simplify terms.

Simplify each term.

Multiply $y$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$3 (y^{2} y) + 3 y \cdot 2 - 2 y^{2} - 2 \cdot 2$

Multiply $y^{2}$ by $y$ .

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $2$ and $1$ .

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

Multiply $2$ by $3$ .

$3 y^{3} + 6 y - 2 y^{2} - 2 \cdot 2$

Multiply $- 2$ by $2$ .

$3 y^{3} + 6 y - 2 y^{2} - 4$

Move $6 y$ .

$3 y^{3} - 2 y^{2} + 6 y - 4$

Simplify (9y^4+14y^2-8)/(3y+2)

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