Factor ab-ac-b^2+2bc-c^2

$a b - a c - b^{2} + 2 b c - c^{2}$

Step 1

Regroup terms.

$a b - a c - b^{2} + 2 b c - c^{2}$

Step 2

Factor $a$ out of $a b - a c$ .

Factor $a$ out of $a b$ .

$a (b) - a c - b^{2} + 2 b c - c^{2}$

Factor $a$ out of $- a c$ .

$a (b) + a (- 1 c) - b^{2} + 2 b c - c^{2}$

Factor $a$ out of $a (b) + a (- 1 c)$ .

$a (b - 1 c) - b^{2} + 2 b c - c^{2}$

Step 3

Rewrite $- 1 c$ as $- c$ .

$a (b - c) - b^{2} + 2 b c - c^{2}$

Step 4

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

Reorder terms.

$a (b - c) - b^{2} - c^{2} + 2 b c$

Reorder $- c^{2}$ and $2 b c$ .

$a (b - c) - b^{2} + 2 b c - c^{2}$

Factor $2$ out of $2 b c$ .

$a (b - c) - b^{2} + 2 (b c) - c^{2}$

Rewrite $2$ as $1$ plus $1$

$a (b - c) - b^{2} + (1 + 1) (b c) - c^{2}$

Apply the distributive property.

$a (b - c) - b^{2} + 1 (b c) + 1 (b c) - c^{2}$

Multiply $b c$ by $1$ .

$a (b - c) - b^{2} + b c + 1 (b c) - c^{2}$

Multiply $b c$ by $1$ .

$a (b - c) - b^{2} + b c + b c - c^{2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$a (b - c) + (- b^{2} + b c) + b c - c^{2}$

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

$a (b - c) + b (- b + c) - c (- b + c)$

Factor the polynomial by factoring out the greatest common factor, $- b + c$ .

$a (b - c) + (- b + c) (b - c)$

Step 5

Factor $b - c$ out of $a (b - c) + (- b + c) (b - c)$ .

Factor $b - c$ out of $a (b - c)$ .

$(b - c) a + (- b + c) (b - c)$

Factor $b - c$ out of $(- b + c) (b - c)$ .

$(b - c) a + (b - c) (- b + c)$

Factor $b - c$ out of $(b - c) a + (b - c) (- b + c)$ .

$(b - c) (a - b + c)$

Factor ab-ac-b^2+2bc-c^2

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