Algebra Examples

$\frac{2 x^{2} \cdot (15 x) + 18}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 1

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 15 x^{2} x + 18}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1

Move $x$ .

$\frac{2 \cdot 15 (x \cdot x^{2}) + 18}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 2.2

Multiply $x$ by $x^{2}$ .

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Step 2.2.1

Raise $x$ to the power of $1$ .

$\frac{2 \cdot 15 (x^{1} x^{2}) + 18}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \cdot 15 x^{1 + 2} + 18}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 2.3

Add $1$ and $2$ .

$\frac{2 \cdot 15 x^{3} + 18}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 3

Multiply $2$ by $15$ .

$\frac{30 x^{3} + 18}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 4

Factor $6$ out of $30 x^{3} + 18$ .

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Step 4.1

Factor $6$ out of $30 x^{3}$ .

$\frac{6 (5 x^{3}) + 18}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 4.2

Factor $6$ out of $18$ .

$\frac{6 (5 x^{3}) + 6 (3)}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 4.3

Factor $6$ out of $6 (5 x^{3}) + 6 (3)$ .

$\frac{6 (5 x^{3} + 3)}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 5

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{6 (5 x^{3} + 3)}{{(2 x)}^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 6

Rewrite $9$ as $3^{2}$ .

$\frac{6 (5 x^{3} + 3)}{{(2 x)}^{2} - 3^{2}} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 3$ .

$\frac{6 (5 x^{3} + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 3 x - 18}$

Step 8

Factor by grouping.

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Step 8.1

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 $a \cdot c = 2 \cdot - 9 = - 18$ and whose sum is $b = 3$ .

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Step 8.1.1

Factor $3$ out of $3 x$ .

$\frac{6 (5 x^{3} + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{2 x^{2} + 3 (x) - 9}{x^{2} + 3 x - 18}$

Step 8.1.2

Rewrite $3$ as $- 3$ plus $6$

$\frac{6 (5 x^{3} + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{2 x^{2} + (- 3 + 6) x - 9}{x^{2} + 3 x - 18}$

Step 8.1.3

Apply the distributive property.

$\frac{6 (5 x^{3} + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{2 x^{2} - 3 x + 6 x - 9}{x^{2} + 3 x - 18}$

Step 8.2

Factor out the greatest common factor from each group.

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Step 8.2.1

Group the first two terms and the last two terms.

$\frac{6 (5 x^{3} + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{(2 x^{2} - 3 x) + 6 x - 9}{x^{2} + 3 x - 18}$

Step 8.2.2

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

$\frac{6 (5 x^{3} + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{x (2 x - 3) + 3 (2 x - 3)}{x^{2} + 3 x - 18}$

Step 8.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$\frac{6 (5 x^{3} + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{(2 x - 3) (x + 3)}{x^{2} + 3 x - 18}$

Step 9

Factor $x^{2} + 3 x - 18$ using the AC method.

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Step 9.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $3$ .

$- 3, 6$

Step 9.2

Write the factored form using these integers.

$\frac{6 (5 x^{3} + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{(2 x - 3) (x + 3)}{(x - 3) (x + 6)}$