Pre-Algebra Examples

$\frac{20 a^{2} + 7 a b - 3 b^{2}}{4 a - b}$

Step 1

Simplify the numerator.

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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 $a \cdot c = 20 \cdot - 3 = - 60$ and whose sum is $b = 7$ .

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Step 1.1.1.1

Reorder terms.

$\frac{20 a^{2} - 3 b^{2} + 7 a b}{4 a - b}$

Step 1.1.1.2

Reorder $- 3 b^{2}$ and $7 a b$ .

$\frac{20 a^{2} + 7 a b - 3 b^{2}}{4 a - b}$

Step 1.1.1.3

Factor $7$ out of $7 a b$ .

$\frac{20 a^{2} + 7 (a b) - 3 b^{2}}{4 a - b}$

Step 1.1.1.4

Rewrite $7$ as $- 5$ plus $12$

$\frac{20 a^{2} + (- 5 + 12) (a b) - 3 b^{2}}{4 a - b}$

Step 1.1.1.5

Apply the distributive property.

$\frac{20 a^{2} - 5 (a b) + 12 (a b) - 3 b^{2}}{4 a - b}$

Step 1.1.1.6

Move parentheses.

$\frac{20 a^{2} - 5 a b + 12 a b - 3 b^{2}}{4 a - b}$

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.

$\frac{(20 a^{2} - 5 a b) + 12 a b - 3 b^{2}}{4 a - b}$

Step 1.1.2.2

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

$\frac{5 a (4 a - 1 b) + 3 b (4 a - b)}{4 a - b}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $4 a - 1 b$ .

$\frac{(4 a - 1 b) (5 a + 3 b)}{4 a - b}$

Step 1.2

Rewrite $- 1 b$ as $- b$ .

$\frac{(4 a - b) (5 a + 3 b)}{4 a - b}$

Step 2

Cancel the common factor of $4 a - b$ .

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Step 2.1

Cancel the common factor.

$\frac{(4 a - b) (5 a + 3 b)}{4 a - b}$

Step 2.2

Divide $5 a + 3 b$ by $1$ .

$5 a + 3 b$