How do you factor a^2 - b^2 - 2a + 1?

Question

How do you factor a^2 - b^2 - 2a + 1. Please show work and explain. Thanks

Also, how do you factor (s-2)^2 - (s-2)^5. Show work again and explain. Thanks

Answer 1

a^2 - b^2 - 2a + 1
= (a^2 - 2a + 1) - b^2
= (a-1)^2 - b^2
This is a difference of squares.
= ((a-1)+b)((a-1)-b)
= (a-1+b)(a-1-b)

(s-2)^2 - (s-2)^5
First factor out the greatest common factor.
(s-2)^2 (1 - (s-2)^3)
(1 - (s-2)^3) is a difference of two cubes.
The formula you need to use is
a^3 - b^3 = (a-b)(a^2 +ab + b^2)
a = 1 and b = s-2
Plug in and simplify

Answer 2

Let p(a) = a^2 - b^2 - 2a + 1

I'm going to rearrange this

p(a) = a^2 - 2a + 1 - b^2

Now, notice the first three terms are a perfect square. We can factor the first three terms.

p(a) = (a - 1)^2 - b^2

Now, notice we have a difference of squares; we can now factor this.

p(a) = (a - 1 - b) (a - 1 + b)

Answer 3

a^2 - b^2 - 2a + 1
= a^2 - 2a + 1- b^2
= (a-1)^2 - b^2
= (a-1+b)(a-1-b)

(s-2)^2 - (s-2)^5
= (s-2)^2 [1-(s-2)^3]
= (s-2)^2 (3-s)[1+(s-2)+(s-2)^2] ...(used a^3-b^3 = (a-b)(a^2+ab +b^2)

Answer 4

(a-1)^2 -b^2 = (a+b-1)(a-b-1)

because a^2 -2a+1 = (a-1)^2

Answer 5

you need to use quatratic formula.

Answer 6

a^2-2a+1-b^2
=(a-1)^2-b^2
=(a-1+b)(a-1-b)

(s-2)^2-(s-2)^5
(s-2)^2[1-(s-2)^3]
=(s-2)^2[1-(s-2)][1+(s-2)+(s-2)^2]
=(s-2)^2[3-s][s-1+s^2-4s+4]
=(s-2)^2[3-s][s^2-3s+3]