Question

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

Answer 1

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)`