Algebra Examples

$\frac{90 x + 2 x^{2} - 4 x^{3}}{3 x^{2} - 5 x - 50}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 x$ out of $90 x + 2 x^{2} - 4 x^{3}$ .

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Step 1.1.1

Factor $2 x$ out of $90 x$ .

$\frac{2 x (45) + 2 x^{2} - 4 x^{3}}{3 x^{2} - 5 x - 50}$

Step 1.1.2

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

$\frac{2 x (45) + 2 x (x) - 4 x^{3}}{3 x^{2} - 5 x - 50}$

Step 1.1.3

Factor $2 x$ out of $- 4 x^{3}$ .

$\frac{2 x (45) + 2 x (x) + 2 x (- 2 x^{2})}{3 x^{2} - 5 x - 50}$

Step 1.1.4

Factor $2 x$ out of $2 x (45) + 2 x (x)$ .

$\frac{2 x (45 + x) + 2 x (- 2 x^{2})}{3 x^{2} - 5 x - 50}$

Step 1.1.5

Factor $2 x$ out of $2 x (45 + x) + 2 x (- 2 x^{2})$ .

$\frac{2 x (45 + x - 2 x^{2})}{3 x^{2} - 5 x - 50}$

Step 1.2

Factor by grouping.

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Step 1.2.1

Reorder terms.

$\frac{2 x (- 2 x^{2} + x + 45)}{3 x^{2} - 5 x - 50}$

Step 1.2.2

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 $a \cdot c = - 2 \cdot 45 = - 90$ and whose sum is $b = 1$ .

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Step 1.2.2.1

Multiply by $1$ .

$\frac{2 x (- 2 x^{2} + 1 x + 45)}{3 x^{2} - 5 x - 50}$

Step 1.2.2.2

Rewrite $1$ as $- 9$ plus $10$

$\frac{2 x (- 2 x^{2} + (- 9 + 10) x + 45)}{3 x^{2} - 5 x - 50}$

Step 1.2.2.3

Apply the distributive property.

$\frac{2 x (- 2 x^{2} - 9 x + 10 x + 45)}{3 x^{2} - 5 x - 50}$

Step 1.2.3

Factor out the greatest common factor from each group.

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Step 1.2.3.1

Group the first two terms and the last two terms.

$\frac{2 x ((- 2 x^{2} - 9 x) + 10 x + 45)}{3 x^{2} - 5 x - 50}$

Step 1.2.3.2

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

$\frac{2 x (x (- 2 x - 9) - 5 (- 2 x - 9))}{3 x^{2} - 5 x - 50}$

Step 1.2.4

Factor the polynomial by factoring out the greatest common factor, $- 2 x - 9$ .

$\frac{2 x ((- 2 x - 9) (x - 5))}{3 x^{2} - 5 x - 50}$

$\frac{2 x (- 2 x - 9) (x - 5)}{3 x^{2} - 5 x - 50}$

Step 2

Factor by grouping.

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Step 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 $a \cdot c = 3 \cdot - 50 = - 150$ and whose sum is $b = - 5$ .

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Step 2.1.1

Factor $- 5$ out of $- 5 x$ .

$\frac{2 x (- 2 x - 9) (x - 5)}{3 x^{2} - 5 (x) - 50}$

Step 2.1.2

Rewrite $- 5$ as $10$ plus $- 15$

$\frac{2 x (- 2 x - 9) (x - 5)}{3 x^{2} + (10 - 15) x - 50}$

Step 2.1.3

Apply the distributive property.

$\frac{2 x (- 2 x - 9) (x - 5)}{3 x^{2} + 10 x - 15 x - 50}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{2 x (- 2 x - 9) (x - 5)}{(3 x^{2} + 10 x) - 15 x - 50}$

Step 2.2.2

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

$\frac{2 x (- 2 x - 9) (x - 5)}{x (3 x + 10) - 5 (3 x + 10)}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 10$ .

$\frac{2 x (- 2 x - 9) (x - 5)}{(3 x + 10) (x - 5)}$

Step 3

Cancel the common factor of $x - 5$ .

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Step 3.1

Cancel the common factor.

$\frac{2 x (- 2 x - 9) (x - 5)}{(3 x + 10) (x - 5)}$

Step 3.2

Rewrite the expression.

$\frac{2 x (- 2 x - 9)}{3 x + 10}$

Step 4

Factor $- 1$ out of $- 2 x$ .

$\frac{2 x (- (2 x) - 9)}{3 x + 10}$

Step 5

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{2 x (- (2 x) - 1 (9))}{3 x + 10}$

Step 6

Factor $- 1$ out of $- (2 x) - 1 (9)$ .

$\frac{2 x (- (2 x + 9))}{3 x + 10}$

Step 7

Simplify the expression.

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Step 7.1

Rewrite $- (2 x + 9)$ as $- 1 (2 x + 9)$ .

$\frac{2 x (- 1 (2 x + 9))}{3 x + 10}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{(2 x) (2 x + 9)}{3 x + 10}$

Step 7.3

Reorder factors in $- \frac{(2 x) (2 x + 9)}{3 x + 10}$ .

$- \frac{2 x (2 x + 9)}{3 x + 10}$