# Factoring help

#### oldblue

##### New member
Just looking for a little help for this problem. I'm sure I'm making more out of it than it is.

Factor the below by grouping.
3x^3-3x^2-x+1

#### chrisr

##### Full Member
You can write "nested" factors.
3 terms have a common value.
Grouping those and examining what you've got in brackets,
you find common terms a second time.

Think it through, what is your first common term, given that x is some number?

Other than x, 1?

#### chrisr

##### Full Member
Yes,
the more practical way to factorise this is to find f(x)=0.
f(0)=1
f(1)=0
therefore (x-1) is a factor.

To find the second factor... (x-1)(ax[sup:rqdp27i3]2[/sup:rqdp27i3]+bx+c)=3x[sup:rqdp27i3]3[/sup:rqdp27i3]-3x[sup:rqdp27i3]2[/sup:rqdp27i3]-x+1
gives a=3, b=0 and c=-1.

(x-1)(3x[sup:rqdp27i3]2[/sup:rqdp27i3]-1) = 3x[sup:rqdp27i3]3[/sup:rqdp27i3]-3x[sup:rqdp27i3]2[/sup:rqdp27i3]-x+1.

As you have three "x" terms and two "3" terms, you can start grouping terms using the "3".
3(x[sup:rqdp27i3]3[/sup:rqdp27i3]-x[sup:rqdp27i3]2[/sup:rqdp27i3])-x+1 and you see that x[sup:rqdp27i3]2[/sup:rqdp27i3] can be taken out also.

Or, nesting.... x(3x[sup:rqdp27i3]2[/sup:rqdp27i3]-3x-1)+1 = x{3x(x-1)-1}+1
=3x[sup:rqdp27i3]2[/sup:rqdp27i3](x-1)-x+1=-3x[sup:rqdp27i3]2[/sup:rqdp27i3](1-x)+(1-x) = (1-x)(1-3x[sup:rqdp27i3]2[/sup:rqdp27i3]).

#### oldblue

##### New member
Thank you. I wasn't sure of myself, but I will practice more!

#### chrisr

##### Full Member
The factors are based on the fact that "zero multiplied by anything equals zero".

When we write quadratics, cubics etc as multiplications, using the values that cause it to be zero,
the answers are embedded in the factors themselves.

x=5 can be written x-5=0...
2(x-5)=0 then x=5 since 2 is not zero....
(x-2)(x-3)=0.... x is 2 or 3 and so on.

I should have added earlier that for ax[sup:3uhuggyh]2[/sup:3uhuggyh]+bx+c, both "a" and "c" are immediately obvious,
because x(ax[sup:3uhuggyh]2[/sup:3uhuggyh]) is 3x[sup:3uhuggyh]3[/sup:3uhuggyh] and (-1)c = 1.

"b" can be discovered in 2 ways by adding together the "x" parts or the "x[sup:3uhuggyh]2[/sup:3uhuggyh]" parts,
....x(bx)-3x[sup:3uhuggyh]2[/sup:3uhuggyh]=-3x[sup:3uhuggyh]2[/sup:3uhuggyh] so b=0, or -bx-x = -x so b=0.

#### stapel

##### Super Moderator
Staff member
oldblue said:
Factor the below by grouping.
3x^3-3x^2-x+1
When you have four terms and nothing comes out of all of them, you should check for factoring pairs. In your case, you have:

. . . . .(3x[sup:3c3fn4x8]3[/sup:3c3fn4x8] - 3x[sup:3c3fn4x8]2[/sup:3c3fn4x8]) - (x - 1)

. . . . .3x[sup:3c3fn4x8]2[/sup:3c3fn4x8](x - 1) - 1(x - 1)

Take the common factor out front:

. . . . .(x - 1)(3x[sup:3c3fn4x8]2[/sup:3c3fn4x8] - 1)

Hope that helps! :wink: