need help with these three factoring problems?

Question

first: u(v - 1) - 2(1 - v)

second: z2 + 2z + 1 - w2

third: u2 - 10u + 9

Answer 1

1) u(v - 1) - 2(1 - v)

The key thing to notice here is how you can *almost* group those terms together, except for the fact that they're not quite identical (but almost identical).

May I present to you .... "the negative one technique".
Whenever you have a difference between two terms (x - y), how the negative one technique works is that you can change that to (y - x) so long as you multiply by -1. Therefore,
(x - y) is the same as -(y - x), or (-1) (y - x).

This is what we want to do above, for u(v - 1) - 2(1 - v); use the negative one technique to make the desired change. And that's what we do.

u(v - 1) - 2 (-(v - 1))

Note that two negatives cancel into a positive,

u(v - 1) + 2(v - 1)

And now we can group them, since they now both contain a (v - 1).

(v - 1) (u + 2)

2) z^2 + 2z + 1 - w^2

The first thing you need to notice is that the first three terms can be factored as a perfect square. In fact, it does factor into this:

(z + 1)(z + 1) - w^2

Which we can change to

(z + 1)^2 - w^2

Notice that this is a difference of squares. Factoring a^2 - b^2 would give us (a - b) (a + b). It's the exact same thing here, except that our square values are different.. So we factor this like so:

[ (z + 1) - w ] [ (z + 1) + w ]

3. u^2 - 10u + 9

This one's straightforward.
(u + ?) (u + ?)

What two numbers multiply to make 9, and add to make -10? The answer to that is -9 and -1. This is what goes in the question marks.

(u - 9) (u - 1)

Answer 2

first: u(v - 1) - 2(1 - v)
u(v - 1) + 2(-1 + v)
(v - 1)(u + 2)

second: z^2 + 2z + 1 - w^2
(z + 1)^2 - w^2
(z + 1 + w)(z + 1 - w)

third: u^2 - 10u + 9
(u - 1)(u - 9)

Answer 3

1. u(v-1) - 2(1-v) = u(v-1) + 2(v-1) [each term has (v-1) factor]
= (v-1)(u+2)

2. z^2 + 2z + 1 - w^2 = (z+1)^2 - w^2 [Difference of squares]
= [(z+1) - w][(z+1)+w]

3. u^2 - 10u +9 = (u-1)(u-9)

Check is left for you.

Answer 4

1st: u(v - 1) - 2(1 - v)
u(v - 1) + 2(v - 1) = (u + 2)(v - 1)

2nd: z² + 2z + 1 - w²
z(z + 2) + (1 - w)(1 + w)

3rd: u² - 10u + 9
(u - 3)(u - 3)
<::>

Answer 5

u(v - 1) - 2(1 - v)
u(v - 1) - 2(-v + 1)
u(v - 1) - 2(-(v - 1))
u(v - 1) + 2(v - 1)
(u + 2)(v - 1)

----------------------------

z^2 + 2z + 1 - w^2
(z^2 + 2z + 1) - w^2
((z + 1)(z + 1)) - w^2
(z + 1)^2 - w^2
ANS:
((z + 1) - w)((z + 1) + w) or (z + 1 - w)(z + 1 + w)

---------------------------------------

u^2 - 10u + 9
(u - 9)(u - 1)

Answer 6

1.u(v-1)+2(v-1)
=(v-1)(u+2)

2.z^2+2z+2-w^2
=(z+1)^2-w^2
=(z+1+w)(z+1-w)

3.(u-9)(u-1)

Answer 7

1-

uv-u-2+2v
v(u+2)-2-u

2-
4z+1-2w
2(2z-w)+1

3-
u(2-10)+9
-8u+9