Basic Math Examples

$\frac{3 y^{2} - 20 y - 7}{y - 4} \cdot \frac{y^{2} - 12 y + 32}{y - 7}$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 3 \cdot - 7 = - 21$ and whose sum is $b = - 20$ .

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Step 1.1.1

Factor $- 20$ out of $- 20 y$ .

$\frac{3 y^{2} - 20 (y) - 7}{y - 4} \cdot \frac{y^{2} - 12 y + 32}{y - 7}$

Step 1.1.2

Rewrite $- 20$ as $1$ plus $- 21$

$\frac{3 y^{2} + (1 - 21) y - 7}{y - 4} \cdot \frac{y^{2} - 12 y + 32}{y - 7}$

Step 1.1.3

Apply the distributive property.

$\frac{3 y^{2} + 1 y - 21 y - 7}{y - 4} \cdot \frac{y^{2} - 12 y + 32}{y - 7}$

Step 1.1.4

Multiply $y$ by $1$ .

$\frac{3 y^{2} + y - 21 y - 7}{y - 4} \cdot \frac{y^{2} - 12 y + 32}{y - 7}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(3 y^{2} + y) - 21 y - 7}{y - 4} \cdot \frac{y^{2} - 12 y + 32}{y - 7}$

Step 1.2.2

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

$\frac{y (3 y + 1) - 7 (3 y + 1)}{y - 4} \cdot \frac{y^{2} - 12 y + 32}{y - 7}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 y + 1$ .

$\frac{(3 y + 1) (y - 7)}{y - 4} \cdot \frac{y^{2} - 12 y + 32}{y - 7}$

Step 2

Factor $y^{2} - 12 y + 32$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $32$ and whose sum is $- 12$ .

$- 8, - 4$

Step 2.2

Write the factored form using these integers.

$\frac{(3 y + 1) (y - 7)}{y - 4} \cdot \frac{(y - 8) (y - 4)}{y - 7}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $y - 7$ .

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Step 3.1.1

Factor $y - 7$ out of $(3 y + 1) (y - 7)$ .

$\frac{(y - 7) (3 y + 1)}{y - 4} \cdot \frac{(y - 8) (y - 4)}{y - 7}$

Step 3.1.2

Cancel the common factor.

$\frac{(y - 7) (3 y + 1)}{y - 4} \cdot \frac{(y - 8) (y - 4)}{y - 7}$

Step 3.1.3

Rewrite the expression.

$\frac{3 y + 1}{y - 4} \cdot ((y - 8) (y - 4))$

Step 3.2

Cancel the common factor of $y - 4$ .

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Step 3.2.1

Factor $y - 4$ out of $(y - 8) (y - 4)$ .

$\frac{3 y + 1}{y - 4} \cdot ((y - 4) (y - 8))$

Step 3.2.2

Cancel the common factor.

$\frac{3 y + 1}{y - 4} \cdot ((y - 4) (y - 8))$

Step 3.2.3

Rewrite the expression.

$(3 y + 1) \cdot (y - 8)$

Step 4

Expand $(3 y + 1) (y - 8)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$3 y (y - 8) + 1 (y - 8)$

Step 4.2

Apply the distributive property.

$3 y \cdot y + 3 y \cdot - 8 + 1 (y - 8)$

Step 4.3

Apply the distributive property.

$3 y \cdot y + 3 y \cdot - 8 + 1 y + 1 \cdot - 8$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

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Step 5.1.1.1

Move $y$ .

$3 (y \cdot y) + 3 y \cdot - 8 + 1 y + 1 \cdot - 8$

Step 5.1.1.2

Multiply $y$ by $y$ .

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

Step 5.1.2

Multiply $- 8$ by $3$ .

$3 y^{2} - 24 y + 1 y + 1 \cdot - 8$

Step 5.1.3

Multiply $y$ by $1$ .

$3 y^{2} - 24 y + y + 1 \cdot - 8$

Step 5.1.4

Multiply $- 8$ by $1$ .

$3 y^{2} - 24 y + y - 8$

Step 5.2

Add $- 24 y$ and $y$ .

$3 y^{2} - 23 y - 8$