Basic Math Examples

$\frac{20 y^{8} + 16 y^{7} - 4 y^{6}}{4 y^{6}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4 y^{6}$ out of $20 y^{8} + 16 y^{7} - 4 y^{6}$ .

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Step 1.1.1

Factor $4 y^{6}$ out of $20 y^{8}$ .

$\frac{4 y^{6} (5 y^{2}) + 16 y^{7} - 4 y^{6}}{4 y^{6}}$

Step 1.1.2

Factor $4 y^{6}$ out of $16 y^{7}$ .

$\frac{4 y^{6} (5 y^{2}) + 4 y^{6} (4 y) - 4 y^{6}}{4 y^{6}}$

Step 1.1.3

Factor $4 y^{6}$ out of $- 4 y^{6}$ .

$\frac{4 y^{6} (5 y^{2}) + 4 y^{6} (4 y) + 4 y^{6} (- 1)}{4 y^{6}}$

Step 1.1.4

Factor $4 y^{6}$ out of $4 y^{6} (5 y^{2}) + 4 y^{6} (4 y)$ .

$\frac{4 y^{6} (5 y^{2} + 4 y) + 4 y^{6} (- 1)}{4 y^{6}}$

Step 1.1.5

Factor $4 y^{6}$ out of $4 y^{6} (5 y^{2} + 4 y) + 4 y^{6} (- 1)$ .

$\frac{4 y^{6} (5 y^{2} + 4 y - 1)}{4 y^{6}}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = 5 \cdot - 1 = - 5$ and whose sum is $b = 4$ .

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Step 1.2.1.1

Factor $4$ out of $4 y$ .

$\frac{4 y^{6} (5 y^{2} + 4 (y) - 1)}{4 y^{6}}$

Step 1.2.1.2

Rewrite $4$ as $- 1$ plus $5$

$\frac{4 y^{6} (5 y^{2} + (- 1 + 5) y - 1)}{4 y^{6}}$

Step 1.2.1.3

Apply the distributive property.

$\frac{4 y^{6} (5 y^{2} - 1 y + 5 y - 1)}{4 y^{6}}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{4 y^{6} ((5 y^{2} - 1 y) + 5 y - 1)}{4 y^{6}}$

Step 1.2.2.2

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

$\frac{4 y^{6} (y (5 y - 1) + 1 (5 y - 1))}{4 y^{6}}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $5 y - 1$ .

$\frac{4 y^{6} ((5 y - 1) (y + 1))}{4 y^{6}}$

$\frac{4 y^{6} (5 y - 1) (y + 1)}{4 y^{6}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $4$ .

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Step 2.1.1

Cancel the common factor.

$\frac{4 y^{6} (5 y - 1) (y + 1)}{4 y^{6}}$

Step 2.1.2

Rewrite the expression.

$\frac{(y^{6} (5 y - 1)) (y + 1)}{y^{6}}$

Step 2.2

Cancel the common factor of $y^{6}$ .

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Step 2.2.1

Cancel the common factor.

$\frac{y^{6} (5 y - 1) (y + 1)}{y^{6}}$

Step 2.2.2

Divide $(5 y - 1) (y + 1)$ by $1$ .

$(5 y - 1) (y + 1)$

Step 3

Expand $(5 y - 1) (y + 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$5 y (y + 1) - 1 (y + 1)$

Step 3.2

Apply the distributive property.

$5 y \cdot y + 5 y \cdot 1 - 1 (y + 1)$

Step 3.3

Apply the distributive property.

$5 y \cdot y + 5 y \cdot 1 - 1 y - 1 \cdot 1$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $y$ by $y$ by adding the exponents.

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Step 4.1.1.1

Move $y$ .

$5 (y \cdot y) + 5 y \cdot 1 - 1 y - 1 \cdot 1$

Step 4.1.1.2

Multiply $y$ by $y$ .

$5 y^{2} + 5 y \cdot 1 - 1 y - 1 \cdot 1$

Step 4.1.2

Multiply $5$ by $1$ .

$5 y^{2} + 5 y - 1 y - 1 \cdot 1$

Step 4.1.3

Rewrite $- 1 y$ as $- y$ .

$5 y^{2} + 5 y - y - 1 \cdot 1$

Step 4.1.4

Multiply $- 1$ by $1$ .

$5 y^{2} + 5 y - y - 1$

Step 4.2

Subtract $y$ from $5 y$ .

$5 y^{2} + 4 y - 1$