Algebra Examples

$\frac{3 x^{2} + 2 x - 21}{- 2 x^{2} - 2 x + 12} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 1

Factor by grouping.

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Step 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 $a \cdot c = 3 \cdot - 21 = - 63$ and whose sum is $b = 2$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{3 x^{2} + 2 (x) - 21}{- 2 x^{2} - 2 x + 12} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 1.1.2

Rewrite $2$ as $- 7$ plus $9$

$\frac{3 x^{2} + (- 7 + 9) x - 21}{- 2 x^{2} - 2 x + 12} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 1.1.3

Apply the distributive property.

$\frac{3 x^{2} - 7 x + 9 x - 21}{- 2 x^{2} - 2 x + 12} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} - 7 x) + 9 x - 21}{- 2 x^{2} - 2 x + 12} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 1.2.2

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

$\frac{x (3 x - 7) + 3 (3 x - 7)}{- 2 x^{2} - 2 x + 12} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 7$ .

$\frac{(3 x - 7) (x + 3)}{- 2 x^{2} - 2 x + 12} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2$ out of $- 2 x^{2} - 2 x + 12$ .

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Step 2.1.1

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

$\frac{(3 x - 7) (x + 3)}{2 (- x^{2}) - 2 x + 12} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.1.2

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

$\frac{(3 x - 7) (x + 3)}{2 (- x^{2}) + 2 (- x) + 12} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.1.3

Factor $2$ out of $12$ .

$\frac{(3 x - 7) (x + 3)}{2 (- x^{2}) + 2 (- x) + 2 (6)} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.1.4

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

$\frac{(3 x - 7) (x + 3)}{2 (- x^{2} - x) + 2 (6)} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.1.5

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

$\frac{(3 x - 7) (x + 3)}{2 (- x^{2} - x + 6)} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.2

Factor by grouping.

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Step 2.2.1

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 $a \cdot c = - 1 \cdot 6 = - 6$ and whose sum is $b = - 1$ .

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Step 2.2.1.1

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

$\frac{(3 x - 7) (x + 3)}{2 (- x^{2} - (x) + 6)} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.2.1.2

Rewrite $- 1$ as $2$ plus $- 3$

$\frac{(3 x - 7) (x + 3)}{2 (- x^{2} + (2 - 3) x + 6)} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.2.1.3

Apply the distributive property.

$\frac{(3 x - 7) (x + 3)}{2 (- x^{2} + 2 x - 3 x + 6)} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

$\frac{(3 x - 7) (x + 3)}{2 ((- x^{2} + 2 x) - 3 x + 6)} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.2.2.2

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

$\frac{(3 x - 7) (x + 3)}{2 (x (- x + 2) + 3 (- x + 2))} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 2.2.3

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

$\frac{(3 x - 7) (x + 3)}{2 ((- x + 2) (x + 3))} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{2 x^{2} + 25 x + 63}{6 x^{2} + 7 x - 49}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 2 \cdot 63 = 126$ and whose sum is $b = 25$ .

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Step 3.1.1

Factor $25$ out of $25 x$ .

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{2 x^{2} + 25 (x) + 63}{6 x^{2} + 7 x - 49}$

Step 3.1.2

Rewrite $25$ as $7$ plus $18$

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{2 x^{2} + (7 + 18) x + 63}{6 x^{2} + 7 x - 49}$

Step 3.1.3

Apply the distributive property.

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{2 x^{2} + 7 x + 18 x + 63}{6 x^{2} + 7 x - 49}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x^{2} + 7 x) + 18 x + 63}{6 x^{2} + 7 x - 49}$

Step 3.2.2

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

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{x (2 x + 7) + 9 (2 x + 7)}{6 x^{2} + 7 x - 49}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 7$ .

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x + 7) (x + 9)}{6 x^{2} + 7 x - 49}$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 6 \cdot - 49 = - 294$ and whose sum is $b = 7$ .

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Step 4.1.1

Factor $7$ out of $7 x$ .

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x + 7) (x + 9)}{6 x^{2} + 7 (x) - 49}$

Step 4.1.2

Rewrite $7$ as $- 14$ plus $21$

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x + 7) (x + 9)}{6 x^{2} + (- 14 + 21) x - 49}$

Step 4.1.3

Apply the distributive property.

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x + 7) (x + 9)}{6 x^{2} - 14 x + 21 x - 49}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x + 7) (x + 9)}{(6 x^{2} - 14 x) + 21 x - 49}$

Step 4.2.2

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

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x + 7) (x + 9)}{2 x (3 x - 7) + 7 (3 x - 7)}$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 7$ .

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x + 7) (x + 9)}{(3 x - 7) (2 x + 7)}$

Step 5

Cancel the common factor of $3 x - 7$ .

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Step 5.1

Cancel the common factor.

$\frac{(3 x - 7) (x + 3)}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x + 7) (x + 9)}{(3 x - 7) (2 x + 7)}$

Step 5.2

Rewrite the expression.

$\frac{x + 3}{2 (- x + 2) (x + 3)} \cdot \frac{(2 x + 7) (x + 9)}{2 x + 7}$

Step 6

Multiply $\frac{x + 3}{2 (- x + 2) (x + 3)}$ by $\frac{(2 x + 7) (x + 9)}{2 x + 7}$ .

$\frac{(x + 3) ((2 x + 7) (x + 9))}{2 (- x + 2) (x + 3) (2 x + 7)}$

Step 7

Cancel the common factor of $x + 3$ .

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Step 7.1

Cancel the common factor.

$\frac{(x + 3) ((2 x + 7) (x + 9))}{2 (- x + 2) (x + 3) (2 x + 7)}$

Step 7.2

Rewrite the expression.

$\frac{(2 x + 7) (x + 9)}{2 (- x + 2) (2 x + 7)}$

Step 8

Cancel the common factor of $2 x + 7$ .

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Step 8.1

Cancel the common factor.

$\frac{(2 x + 7) (x + 9)}{2 (- x + 2) (2 x + 7)}$

Step 8.2

Rewrite the expression.

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

Step 9

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

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

Step 10

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 11

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

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

Step 12

Rewrite negatives.

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Step 12.1

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

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

Step 12.2

Move the negative in front of the fraction.

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