Algebra Examples

$\frac{3 x^{2} - 7 x + 2}{x^{2} - 4 x + 4}$

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

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Step 1.1.1

Factor $- 7$ out of $- 7 x$ .

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

Step 1.1.2

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

$\frac{3 x^{2} + (- 1 - 6) x + 2}{x^{2} - 4 x + 4}$

Step 1.1.3

Apply the distributive property.

$\frac{3 x^{2} - 1 x - 6 x + 2}{x^{2} - 4 x + 4}$

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.

$\frac{(3 x^{2} - 1 x) - 6 x + 2}{x^{2} - 4 x + 4}$

Step 1.2.2

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

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

Step 1.3

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

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

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 3

Cancel the common factor of $x - 2$ and ${(x - 2)}^{2}$ .

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Step 3.1

Factor $x - 2$ out of $(3 x - 1) (x - 2)$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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