Algebra Examples

$\frac{3 x^{2} - 2 x - 21}{2 x^{2} - 9 x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

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 $- 2 x$ .

$\frac{3 x^{2} - 2 (x) - 21}{2 x^{2} - 9 x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

Step 1.1.2

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

$\frac{3 x^{2} + (7 - 9) x - 21}{2 x^{2} - 9 x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

Step 1.1.3

Apply the distributive property.

$\frac{3 x^{2} + 7 x - 9 x - 21}{2 x^{2} - 9 x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

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.

$\frac{(3 x^{2} + 7 x) - 9 x - 21}{2 x^{2} - 9 x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

Step 1.2.2

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

$\frac{x (3 x + 7) - 3 (3 x + 7)}{2 x^{2} - 9 x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

Step 1.3

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

$\frac{(3 x + 7) (x - 3)}{2 x^{2} - 9 x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

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 $a \cdot c = 2 \cdot 7 = 14$ and whose sum is $b = - 9$ .

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Step 2.1.1

Factor $- 9$ out of $- 9 x$ .

$\frac{(3 x + 7) (x - 3)}{2 x^{2} - 9 (x) + 7} \cdot \frac{2 x - 7}{4 x - 12}$

Step 2.1.2

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

$\frac{(3 x + 7) (x - 3)}{2 x^{2} + (- 2 - 7) x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

Step 2.1.3

Apply the distributive property.

$\frac{(3 x + 7) (x - 3)}{2 x^{2} - 2 x - 7 x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

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.

$\frac{(3 x + 7) (x - 3)}{(2 x^{2} - 2 x) - 7 x + 7} \cdot \frac{2 x - 7}{4 x - 12}$

Step 2.2.2

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

$\frac{(3 x + 7) (x - 3)}{2 x (x - 1) - 7 (x - 1)} \cdot \frac{2 x - 7}{4 x - 12}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$\frac{(3 x + 7) (x - 3)}{(x - 1) (2 x - 7)} \cdot \frac{2 x - 7}{4 x - 12}$

Step 3

Factor $4$ out of $4 x - 12$ .

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Step 3.1

Factor $4$ out of $4 x$ .

$\frac{(3 x + 7) (x - 3)}{(x - 1) (2 x - 7)} \cdot \frac{2 x - 7}{4 (x) - 12}$

Step 3.2

Factor $4$ out of $- 12$ .

$\frac{(3 x + 7) (x - 3)}{(x - 1) (2 x - 7)} \cdot \frac{2 x - 7}{4 x + 4 \cdot - 3}$

Step 3.3

Factor $4$ out of $4 x + 4 \cdot - 3$ .

$\frac{(3 x + 7) (x - 3)}{(x - 1) (2 x - 7)} \cdot \frac{2 x - 7}{4 (x - 3)}$

Step 4

Cancel the common factor of $x - 3$ .

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Step 4.1

Factor $x - 3$ out of $(3 x + 7) (x - 3)$ .

$\frac{(x - 3) (3 x + 7)}{(x - 1) (2 x - 7)} \cdot \frac{2 x - 7}{4 (x - 3)}$

Step 4.2

Factor $x - 3$ out of $4 (x - 3)$ .

$\frac{(x - 3) (3 x + 7)}{(x - 1) (2 x - 7)} \cdot \frac{2 x - 7}{(x - 3) \cdot 4}$

Step 4.3

Cancel the common factor.

$\frac{(x - 3) (3 x + 7)}{(x - 1) (2 x - 7)} \cdot \frac{2 x - 7}{(x - 3) \cdot 4}$

Step 4.4

Rewrite the expression.

$\frac{3 x + 7}{(x - 1) (2 x - 7)} \cdot \frac{2 x - 7}{4}$

Step 5

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

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Step 5.1

Factor $2 x - 7$ out of $(x - 1) (2 x - 7)$ .

$\frac{3 x + 7}{(2 x - 7) (x - 1)} \cdot \frac{2 x - 7}{4}$

Step 5.2

Cancel the common factor.

$\frac{3 x + 7}{(2 x - 7) (x - 1)} \cdot \frac{2 x - 7}{4}$

Step 5.3

Rewrite the expression.

$\frac{3 x + 7}{x - 1} \cdot \frac{1}{4}$

Step 6

Multiply $\frac{3 x + 7}{x - 1}$ by $\frac{1}{4}$ .

$\frac{3 x + 7}{(x - 1) \cdot 4}$

Step 7

Move $4$ to the left of $x - 1$ .

$\frac{3 x + 7}{4 (x - 1)}$