Square of a Binomial

The square of a binomial is always a trinomial.  It will be helpful to memorize these patterns for writing squares of binomials as trinomials.

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2

Examples:

Square each binomial.

a) (x + 4)2

    (x + 4)2 = x2 + 2(x · 4) + 42

                 = x2 + 8x + 16

b) (2y – 3)2

    (2y – 3)2 = (2y)2 – 2(2y · 3) + 32

                  = (2y)2 – 2(6y) + 32

                  = 4y2– 12y + 9  

c) (3p – 2q2)

    (3p – 2q2) = (3p)2 – 2(3p · 2q2) + (2q2)2

                     = 9p2– 2(6pq2) + 4q4

                     = 9p2– 12pq2 + 4q4

If the coefficients of a trinomial ax2 + bx + c satisfy the equation

then the trinomial is the perfect square of the binomial

Example 1:

Factor, if possible.

x2 – 14x + 49

Here, a = 1, b= 14, and c = 49. We have:

So, the trinomial is a perfect square:

You can verify this using FOIL.

Example 2:

Factor, if possible.

9w4 + 12w2 + 4

Here, a = 9, b = 12, and c = 4. (We can treat w2 as x, and not worry about the fourth power.)

So, the trinomial is a perfect square:

This can also be verified using FOIL.