Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Factor by grouping.

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Step 1.3.1

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 $b = 14$ .

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Step 1.3.1.1

Factor $14$ out of $14 u$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Apply the distributive property.

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

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

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

Step 1.3.2.2

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

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

Step 1.3.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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 $3 y + 2$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

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.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

Move $y^{2}$ .

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

Step 4.1.1.2

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

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Step 4.1.1.2.1

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

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

Step 4.1.1.2.2

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

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

Step 4.1.1.3

Add $2$ and $1$ .

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

Step 4.1.2

Multiply $2$ by $3$ .

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

Step 4.1.3

Multiply $- 2$ by $2$ .

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

Step 4.2

Move $6 y$ .

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