Pre-Algebra Examples

$\frac{2}{3 x^{2} - 20 x + 32} + \frac{1}{3 x^{2} - 14 x + 16}$

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 $a \cdot c = 3 \cdot 32 = 96$ and whose sum is $b = - 20$ .

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Step 1.1.1.1

Factor $- 20$ out of $- 20 x$ .

$\frac{2}{3 x^{2} - 20 (x) + 32} + \frac{1}{3 x^{2} - 14 x + 16}$

Step 1.1.1.2

Rewrite $- 20$ as $- 8$ plus $- 12$

$\frac{2}{3 x^{2} + (- 8 - 12) x + 32} + \frac{1}{3 x^{2} - 14 x + 16}$

Step 1.1.1.3

Apply the distributive property.

$\frac{2}{3 x^{2} - 8 x - 12 x + 32} + \frac{1}{3 x^{2} - 14 x + 16}$

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.

$\frac{2}{(3 x^{2} - 8 x) - 12 x + 32} + \frac{1}{3 x^{2} - 14 x + 16}$

Step 1.1.2.2

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

$\frac{2}{x (3 x - 8) - 4 (3 x - 8)} + \frac{1}{3 x^{2} - 14 x + 16}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 8$ .

$\frac{2}{(3 x - 8) (x - 4)} + \frac{1}{3 x^{2} - 14 x + 16}$

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 $a \cdot c = 3 \cdot 16 = 48$ and whose sum is $b = - 14$ .

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Step 1.2.1.1

Factor $- 14$ out of $- 14 x$ .

$\frac{2}{(3 x - 8) (x - 4)} + \frac{1}{3 x^{2} - 14 (x) + 16}$

Step 1.2.1.2

Rewrite $- 14$ as $- 6$ plus $- 8$

$\frac{2}{(3 x - 8) (x - 4)} + \frac{1}{3 x^{2} + (- 6 - 8) x + 16}$

Step 1.2.1.3

Apply the distributive property.

$\frac{2}{(3 x - 8) (x - 4)} + \frac{1}{3 x^{2} - 6 x - 8 x + 16}$

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.

$\frac{2}{(3 x - 8) (x - 4)} + \frac{1}{(3 x^{2} - 6 x) - 8 x + 16}$

Step 1.2.2.2

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

$\frac{2}{(3 x - 8) (x - 4)} + \frac{1}{3 x (x - 2) - 8 (x - 2)}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$\frac{2}{(3 x - 8) (x - 4)} + \frac{1}{(x - 2) (3 x - 8)}$

Step 2

To write $\frac{2}{(3 x - 8) (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{2}{(3 x - 8) (x - 4)} \cdot \frac{x - 2}{x - 2} + \frac{1}{(x - 2) (3 x - 8)}$

Step 3

To write $\frac{1}{(x - 2) (3 x - 8)}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{2}{(3 x - 8) (x - 4)} \cdot \frac{x - 2}{x - 2} + \frac{1}{(x - 2) (3 x - 8)} \cdot \frac{x - 4}{x - 4}$

Step 4

Write each expression with a common denominator of $(3 x - 8) (x - 4) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{(3 x - 8) (x - 4)}$ by $\frac{x - 2}{x - 2}$ .

$\frac{2 (x - 2)}{(3 x - 8) (x - 4) (x - 2)} + \frac{1}{(x - 2) (3 x - 8)} \cdot \frac{x - 4}{x - 4}$

Step 4.2

Multiply $\frac{1}{(x - 2) (3 x - 8)}$ by $\frac{x - 4}{x - 4}$ .

$\frac{2 (x - 2)}{(3 x - 8) (x - 4) (x - 2)} + \frac{x - 4}{(x - 2) (3 x - 8) (x - 4)}$

Step 4.3

Reorder the factors of $(3 x - 8) (x - 4) (x - 2)$ .

$\frac{2 (x - 2)}{(3 x - 8) (x - 2) (x - 4)} + \frac{x - 4}{(x - 2) (3 x - 8) (x - 4)}$

Step 4.4

Reorder the factors of $(x - 2) (3 x - 8) (x - 4)$ .

$\frac{2 (x - 2)}{(3 x - 8) (x - 2) (x - 4)} + \frac{x - 4}{(3 x - 8) (x - 2) (x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 (x - 2) + x - 4}{(3 x - 8) (x - 2) (x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 x + 2 \cdot - 2 + x - 4}{(3 x - 8) (x - 2) (x - 4)}$

Step 6.2

Multiply $2$ by $- 2$ .

$\frac{2 x - 4 + x - 4}{(3 x - 8) (x - 2) (x - 4)}$

Step 6.3

Add $2 x$ and $x$ .

$\frac{3 x - 4 - 4}{(3 x - 8) (x - 2) (x - 4)}$

Step 6.4

Subtract $4$ from $- 4$ .

$\frac{3 x - 8}{(3 x - 8) (x - 2) (x - 4)}$

Step 7

Cancel the common factor of $3 x - 8$ .

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Step 7.1

Cancel the common factor.

$\frac{3 x - 8}{(3 x - 8) (x - 2) (x - 4)}$

Step 7.2

Rewrite the expression.

$\frac{1}{(x - 2) (x - 4)}$