Algebra Examples

$\frac{12 x^{3} - 14 x^{2} - 40 x}{2 x - 5}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 x$ out of $12 x^{3} - 14 x^{2} - 40 x$ .

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Step 1.1.1

Factor $2 x$ out of $12 x^{3}$ .

$\frac{2 x (6 x^{2}) - 14 x^{2} - 40 x}{2 x - 5}$

Step 1.1.2

Factor $2 x$ out of $- 14 x^{2}$ .

$\frac{2 x (6 x^{2}) + 2 x (- 7 x) - 40 x}{2 x - 5}$

Step 1.1.3

Factor $2 x$ out of $- 40 x$ .

$\frac{2 x (6 x^{2}) + 2 x (- 7 x) + 2 x (- 20)}{2 x - 5}$

Step 1.1.4

Factor $2 x$ out of $2 x (6 x^{2}) + 2 x (- 7 x)$ .

$\frac{2 x (6 x^{2} - 7 x) + 2 x (- 20)}{2 x - 5}$

Step 1.1.5

Factor $2 x$ out of $2 x (6 x^{2} - 7 x) + 2 x (- 20)$ .

$\frac{2 x (6 x^{2} - 7 x - 20)}{2 x - 5}$

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 $a \cdot c = 6 \cdot - 20 = - 120$ and whose sum is $b = - 7$ .

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Step 1.2.1.1

Factor $- 7$ out of $- 7 x$ .

$\frac{2 x (6 x^{2} - 7 (x) - 20)}{2 x - 5}$

Step 1.2.1.2

Rewrite $- 7$ as $8$ plus $- 15$

$\frac{2 x (6 x^{2} + (8 - 15) x - 20)}{2 x - 5}$

Step 1.2.1.3

Apply the distributive property.

$\frac{2 x (6 x^{2} + 8 x - 15 x - 20)}{2 x - 5}$

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.

$\frac{2 x ((6 x^{2} + 8 x) - 15 x - 20)}{2 x - 5}$

Step 1.2.2.2

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

$\frac{2 x (2 x (3 x + 4) - 5 (3 x + 4))}{2 x - 5}$

Step 1.2.3

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

$\frac{2 x ((3 x + 4) (2 x - 5))}{2 x - 5}$

$\frac{2 x (3 x + 4) (2 x - 5)}{2 x - 5}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $2 x - 5$ .

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Step 2.1.1

Cancel the common factor.

$\frac{2 x (3 x + 4) (2 x - 5)}{2 x - 5}$

Step 2.1.2

Divide $2 x (3 x + 4)$ by $1$ .

$2 x (3 x + 4)$

Step 2.2

Simplify by multiplying through.

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Step 2.2.1

Apply the distributive property.

$2 x (3 x) + 2 x \cdot 4$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Rewrite using the commutative property of multiplication.

$2 \cdot 3 x \cdot x + 2 x \cdot 4$

Step 2.2.2.2

Multiply $4$ by $2$ .

$2 \cdot 3 x \cdot x + 8 x$

Step 2.3

Simplify each term.

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Step 2.3.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.3.1.1

Move $x$ .

$2 \cdot 3 (x \cdot x) + 8 x$

Step 2.3.1.2

Multiply $x$ by $x$ .

$2 \cdot 3 x^{2} + 8 x$

Step 2.3.2

Multiply $2$ by $3$ .

$6 x^{2} + 8 x$