Pre-Algebra Examples

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

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

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

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = 3 \cdot - 8 = - 24

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

b = - 2

$b = - 2$ .

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Step 1.1.1.1

Factor

- 2

$- 2$ out of

- 2 y

$- 2 y$ .

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

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

Step 1.1.1.2

Rewrite

- 2

$- 2$ as

4

$4$ plus

- 6

$- 6$

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

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

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}

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

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

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

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.

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