How to creates Factor by Substitution ?
Can you factor this polynomial?
x2 + 3x + 2
(For this tutorial, I'm going to assume that you know how to do some basic factoring.) The polynomial above factors like this:
x2 + 3x + 2 = (x + 2)(x + 1).
Now, it factors the same way no matter what the variable is, right?
y2 + 3y + 2 = (y + 2)(y + 1).
In fact, you could have any variable, expression or number substituted in place of x and it would still work:
(qr - s )2 + 3(qr - s) + 2 = ((qr -s )+2)((qr -s) + 1)
Look! We've managed to factor that really nasty-looking stuff on the left, without much effort at all! Hint: If you're getting confused, it might help to make up a new variable for the expression. For example, if you want to factor.
(x + 1)2 - 8(x + 1) + 16,
you can make up a new variable A to represent the expression (x + 1). Then rewrite your problem using the new variable, and you can see that it's easy to factor.
A2 - 8A + 16 (Rewrite using the variable A)
(A - 4)2 (....it factors as a perfect square.)
Then you can substitute (x + 1) back in place of A.
((x + 1) - 4)2
= (x -3)2