Simplify ((3x^2-2x-1)/(x^2-3x+2))÷((3x^2+7x+2)/(3x^2+5x-2))

$\frac{3 x^{2} - 2 x - 1}{x^{2} - 3 x + 2} \div \frac{3 x^{2} + 7 x + 2}{3 x^{2} + 5 x - 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} - 2 x - 1}{x^{2} - 3 x + 2} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

Step 2

Factor by grouping.

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 $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

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

$\frac{3 x^{2} - 2 (x) - 1}{x^{2} - 3 x + 2} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

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

$\frac{3 x^{2} + (1 - 3) x - 1}{x^{2} - 3 x + 2} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

Apply the distributive property.

$\frac{3 x^{2} + 1 x - 3 x - 1}{x^{2} - 3 x + 2} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

Multiply $x$ by $1$ .

$\frac{3 x^{2} + x - 3 x - 1}{x^{2} - 3 x + 2} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + x) - 3 x - 1}{x^{2} - 3 x + 2} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

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

$\frac{x (3 x + 1) - (3 x + 1)}{x^{2} - 3 x + 2} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$\frac{(3 x + 1) (x - 1)}{x^{2} - 3 x + 2} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

Step 3

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

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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$\frac{(3 x + 1) (x - 1)}{(x - 2) (x - 1)} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

Step 4

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{(3 x + 1) (x - 1)}{(x - 2) (x - 1)} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

Rewrite the expression.

$\frac{3 x + 1}{x - 2} \cdot \frac{3 x^{2} + 5 x - 2}{3 x^{2} + 7 x + 2}$

Step 5

Factor by grouping.

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$\frac{3 x + 1}{x - 2} \cdot \frac{3 x^{2} + 5 (x) - 2}{3 x^{2} + 7 x + 2}$

Rewrite $5$ as $- 1$ plus $6$

$\frac{3 x + 1}{x - 2} \cdot \frac{3 x^{2} + (- 1 + 6) x - 2}{3 x^{2} + 7 x + 2}$

Apply the distributive property.

$\frac{3 x + 1}{x - 2} \cdot \frac{3 x^{2} - 1 x + 6 x - 2}{3 x^{2} + 7 x + 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{3 x + 1}{x - 2} \cdot \frac{(3 x^{2} - 1 x) + 6 x - 2}{3 x^{2} + 7 x + 2}$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

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

Step 6

Factor by grouping.

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 $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = 7$ .

Factor $7$ out of $7 x$ .

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

Rewrite $7$ as $1$ plus $6$

$\frac{3 x + 1}{x - 2} \cdot \frac{(3 x - 1) (x + 2)}{3 x^{2} + (1 + 6) x + 2}$

Apply the distributive property.

$\frac{3 x + 1}{x - 2} \cdot \frac{(3 x - 1) (x + 2)}{3 x^{2} + 1 x + 6 x + 2}$

Multiply $x$ by $1$ .

$\frac{3 x + 1}{x - 2} \cdot \frac{(3 x - 1) (x + 2)}{3 x^{2} + x + 6 x + 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{3 x + 1}{x - 2} \cdot \frac{(3 x - 1) (x + 2)}{(3 x^{2} + x) + 6 x + 2}$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

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

Step 7

Cancel the common factor of $3 x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 8

Multiply $\frac{1}{x - 2}$ by $\frac{(3 x - 1) (x + 2)}{x + 2}$ .

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

Step 9

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{3 x - 1}{x - 2}$

Simplify ((3x^2-2x-1)/(x^2-3x+2))÷((3x^2+7x+2)/(3x^2+5x-2))

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