Pre-Algebra Examples

$\frac{9 y + 2}{3 y^{2} - 2 y - 8} + \frac{7}{3 y^{2} + y - 4}$

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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 $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = - 2$ .

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Step 1.1.1.1

Factor $- 2$ out of $- 2 y$ .

$\frac{9 y + 2}{3 y^{2} - 2 (y) - 8} + \frac{7}{3 y^{2} + y - 4}$

Step 1.1.1.2

Rewrite $- 2$ as $4$ plus $- 6$

$\frac{9 y + 2}{3 y^{2} + (4 - 6) y - 8} + \frac{7}{3 y^{2} + y - 4}$

Step 1.1.1.3

Apply the distributive property.

$\frac{9 y + 2}{3 y^{2} + 4 y - 6 y - 8} + \frac{7}{3 y^{2} + y - 4}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{9 y + 2}{(3 y^{2} + 4 y) - 6 y - 8} + \frac{7}{3 y^{2} + y - 4}$

Step 1.1.2.2

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

$\frac{9 y + 2}{y (3 y + 4) - 2 (3 y + 4)} + \frac{7}{3 y^{2} + y - 4}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $3 y + 4$ .

$\frac{9 y + 2}{(3 y + 4) (y - 2)} + \frac{7}{3 y^{2} + y - 4}$

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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

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Step 1.2.1.1

Multiply by $1$ .

$\frac{9 y + 2}{(3 y + 4) (y - 2)} + \frac{7}{3 y^{2} + 1 y - 4}$

Step 1.2.1.2

Rewrite $1$ as $- 3$ plus $4$

$\frac{9 y + 2}{(3 y + 4) (y - 2)} + \frac{7}{3 y^{2} + (- 3 + 4) y - 4}$

Step 1.2.1.3

Apply the distributive property.

$\frac{9 y + 2}{(3 y + 4) (y - 2)} + \frac{7}{3 y^{2} - 3 y + 4 y - 4}$

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.

$\frac{9 y + 2}{(3 y + 4) (y - 2)} + \frac{7}{(3 y^{2} - 3 y) + 4 y - 4}$

Step 1.2.2.2

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

$\frac{9 y + 2}{(3 y + 4) (y - 2)} + \frac{7}{3 y (y - 1) + 4 (y - 1)}$

Step 1.2.3

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

$\frac{9 y + 2}{(3 y + 4) (y - 2)} + \frac{7}{(y - 1) (3 y + 4)}$

Step 2

To write $\frac{9 y + 2}{(3 y + 4) (y - 2)}$ as a fraction with a common denominator, multiply by $\frac{y - 1}{y - 1}$ .

$\frac{9 y + 2}{(3 y + 4) (y - 2)} \cdot \frac{y - 1}{y - 1} + \frac{7}{(y - 1) (3 y + 4)}$

Step 3

To write $\frac{7}{(y - 1) (3 y + 4)}$ as a fraction with a common denominator, multiply by $\frac{y - 2}{y - 2}$ .

$\frac{9 y + 2}{(3 y + 4) (y - 2)} \cdot \frac{y - 1}{y - 1} + \frac{7}{(y - 1) (3 y + 4)} \cdot \frac{y - 2}{y - 2}$

Step 4

Write each expression with a common denominator of $(3 y + 4) (y - 2) (y - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{9 y + 2}{(3 y + 4) (y - 2)}$ by $\frac{y - 1}{y - 1}$ .

$\frac{(9 y + 2) (y - 1)}{(3 y + 4) (y - 2) (y - 1)} + \frac{7}{(y - 1) (3 y + 4)} \cdot \frac{y - 2}{y - 2}$

Step 4.2

Multiply $\frac{7}{(y - 1) (3 y + 4)}$ by $\frac{y - 2}{y - 2}$ .

$\frac{(9 y + 2) (y - 1)}{(3 y + 4) (y - 2) (y - 1)} + \frac{7 (y - 2)}{(y - 1) (3 y + 4) (y - 2)}$

Step 4.3

Reorder the factors of $(3 y + 4) (y - 2) (y - 1)$ .

$\frac{(9 y + 2) (y - 1)}{(3 y + 4) (y - 1) (y - 2)} + \frac{7 (y - 2)}{(y - 1) (3 y + 4) (y - 2)}$

Step 4.4

Reorder the factors of $(y - 1) (3 y + 4) (y - 2)$ .

$\frac{(9 y + 2) (y - 1)}{(3 y + 4) (y - 1) (y - 2)} + \frac{7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(9 y + 2) (y - 1) + 7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(9 y + 2) (y - 1)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{9 y (y - 1) + 2 (y - 1) + 7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.1.2

Apply the distributive property.

$\frac{9 y \cdot y + 9 y \cdot - 1 + 2 (y - 1) + 7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.1.3

Apply the distributive property.

$\frac{9 y \cdot y + 9 y \cdot - 1 + 2 y + 2 \cdot - 1 + 7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

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Step 6.2.1.1.1

Move $y$ .

$\frac{9 (y \cdot y) + 9 y \cdot - 1 + 2 y + 2 \cdot - 1 + 7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.2.1.1.2

Multiply $y$ by $y$ .

$\frac{9 y^{2} + 9 y \cdot - 1 + 2 y + 2 \cdot - 1 + 7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.2.1.2

Multiply $- 1$ by $9$ .

$\frac{9 y^{2} - 9 y + 2 y + 2 \cdot - 1 + 7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.2.1.3

Multiply $2$ by $- 1$ .

$\frac{9 y^{2} - 9 y + 2 y - 2 + 7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.2.2

Add $- 9 y$ and $2 y$ .

$\frac{9 y^{2} - 7 y - 2 + 7 (y - 2)}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.3

Apply the distributive property.

$\frac{9 y^{2} - 7 y - 2 + 7 y + 7 \cdot - 2}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.4

Multiply $7$ by $- 2$ .

$\frac{9 y^{2} - 7 y - 2 + 7 y - 14}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.5

Add $- 7 y$ and $7 y$ .

$\frac{9 y^{2} + 0 - 2 - 14}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.6

Add $9 y^{2}$ and $0$ .

$\frac{9 y^{2} - 2 - 14}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.7

Subtract $14$ from $- 2$ .

$\frac{9 y^{2} - 16}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.8

Rewrite $9 y^{2} - 16$ in a factored form.

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Step 6.8.1

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

$\frac{{(3 y)}^{2} - 16}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.8.2

Rewrite $16$ as $4^{2}$ .

$\frac{{(3 y)}^{2} - 4^{2}}{(3 y + 4) (y - 1) (y - 2)}$

Step 6.8.3

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 $b = 4$ .

$\frac{(3 y + 4) (3 y - 4)}{(3 y + 4) (y - 1) (y - 2)}$

Step 7

Cancel the common factor of $3 y + 4$ .

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Step 7.1

Cancel the common factor.

$\frac{(3 y + 4) (3 y - 4)}{(3 y + 4) (y - 1) (y - 2)}$

Step 7.2

Rewrite the expression.

$\frac{3 y - 4}{(y - 1) (y - 2)}$