Basic Math Examples

$\frac{8 y^{2} + 1}{4 y^{2} + 2 y - 6}$

Step 1

Factor $2$ out of $4 y^{2} + 2 y - 6$ .

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Step 1.1

Factor $2$ out of $4 y^{2}$ .

$\frac{8 y^{2} + 1}{2 (2 y^{2}) + 2 y - 6}$

Step 1.2

Factor $2$ out of $2 y$ .

$\frac{8 y^{2} + 1}{2 (2 y^{2}) + 2 (y) - 6}$

Step 1.3

Factor $2$ out of $- 6$ .

$\frac{8 y^{2} + 1}{2 (2 y^{2}) + 2 y + 2 \cdot - 3}$

Step 1.4

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

$\frac{8 y^{2} + 1}{2 (2 y^{2} + y) + 2 \cdot - 3}$

Step 1.5

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

$\frac{8 y^{2} + 1}{2 (2 y^{2} + y - 3)}$

Step 2

Factor by grouping.

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Step 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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 1$ .

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Step 2.1.1

Multiply by $1$ .

$\frac{8 y^{2} + 1}{2 (2 y^{2} + 1 y - 3)}$

Step 2.1.2

Rewrite $1$ as $- 2$ plus $3$

$\frac{8 y^{2} + 1}{2 (2 y^{2} + (- 2 + 3) y - 3)}$

Step 2.1.3

Apply the distributive property.

$\frac{8 y^{2} + 1}{2 (2 y^{2} - 2 y + 3 y - 3)}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{8 y^{2} + 1}{2 ((2 y^{2} - 2 y) + 3 y - 3)}$

Step 2.2.2

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

$\frac{8 y^{2} + 1}{2 (2 y (y - 1) + 3 (y - 1))}$

Step 2.3

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

$\frac{8 y^{2} + 1}{2 ((y - 1) (2 y + 3))}$

$\frac{8 y^{2} + 1}{2 (y - 1) (2 y + 3)}$