Is #100 - 20y + y^2# a perfect square trinomial and how do you factor it?

1 Answer
Jul 26, 2015

Answer:

Yes, your expression is a perfect square trinomial.

Explanation:

A trinomial is simply a polynomial that has three terms. Your expression is a trinomial because it has three terms: #y^2#, #-20y#, and #100#.

In order for it to be a perfect square trinomial, it has to be equal to the square of a binomial, which is a polynomial that has two terms.

You can factor your expression in order to check if it meets this criterion. A trinomial can be factored as a product of two binomials, so all you have to do is see if these two binomials are equal

#y^2 - 20y + 100 = (y+a)(y+b)#

The right side of this equation is equal to

#(y+a) * (y+b) = y^2 + a*y + b* y + b^2#

#(y+a)(y+b) = y^2 + (a+b)y + b^2#

Now match the terms of your original expression with these terms

#y^2 color(blue)(-20)y + color(green)(100) = y^2 + color(blue)((a+b))y + color(green)(b^2)#

You need to have

#b^2 = 100 => b = +-10#

#(a+b) = -20#

  • If #b=10#, then

#a = -20-10 = -30#

This solution is not valid because the expression would be equal to

#(y-30) * (y+10) = color(red)(cancel(color(black)(y^2 -20y - 300)))#

  • If #b=-10#, then

#a = -20 + 10 = -10#

This is a valid solution, since

#(y-10) * (y-10) = y^2 - 20y + 100#

Therefore, your original expression is a perfect square trinomial because you can write it as

#y^2 - 20y + 100 = (y-10) * (y-10) = color(green)((y-10)^2)#