Trinomials Squares of Binomials

Date: 04/10/2003 at 22:40:32
From: Rickey
Subject: Finding out which trinomials are squares of binomials

I keep getting answers in forums about this subject, but no one is 
teaching me how. 

Example: x squared -2x + 1... 

I have to figure out if that trinomial is the square of a binomial.
I factored it out but then couldn't do anything.

Date: 04/10/2003 at 23:15:52
From: Doctor Peterson
Subject: Re: Finding out which trinomials are squares of binomials

Hi, Rickey.

All you do is compare it to the general formula

    (a+b)^2 = a^2 + 2ab + b^2

So IF a trinomial is a square, then the binomial of which it is the 
square can be found by taking the square roots of the first and last 
terms, and adding them together. Then you can check whether it IS 
the square of this binomial.

In your case, compare

    x^2 - 2x + 1

to

    a^2 + 2ab + b^2

We have to have

    a^2 = x^2  so  a = x

and

    b^2 = 1    so  b = 1

But there's one more trick: there are TWO square roots of any number, 
one positive and one negative; so we have to choose what sign to use. 
We can just take the positive for a, and use "x"; then for b, we can 
use the sign of the middle term, which in this case is negative. Why? 
Because when we multiply to get 2ab, if a is positive, we can only get 
a negative result if b is negative.

So what do we get when we square our binomial,

    (a + b) = (x + -1) ?

We get

    (x - 1)^2 = x^2 - 2x + 1

which is just what we wanted.

Do you see how this works? We first choose a and b, and then check 
whether the other term actually works.

You will soon learn to "complete the square"; then you will be looking 
at the first two terms, rather than the first and last, and doing the 
same sort of thing inside out. But for now, I think this approach is 
easier.

You mentioned factoring. If you can do that, then you don't really 
need to bother with any of this. Just look at your two factors, and 
see whether they are the same. If so, then you are squaring a binomial 
to get your trinomial. It's that simple.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/