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McGraw Hill Integrated II, 2012
MH
McGraw Hill Integrated II, 2012 View details
8. Differences of Squares
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Exercise 56 Page 62

Practice makes perfect
a

Let's complete the given table.

Polynomial Factored Polynomial First Term Last Term Middle Term
4x^2+12x+9 (2x+3)(2x+3) 4x^2=(2x)^2 9=3^2 12x
9x^2-24x+16 (3x-4)(3x-4) 9x^2=(3x)^2 16=4^2 - 24x
4x^2-20x+25 (2x-5)(2x-5) 4x^2=(2x)^2 25=5^2 - 20x
16x^2+24x+9 (4x+3)(4x+3) 16x^2=(4x)^2 9=3^2 24x
25x^2+20x+4 (5x+2)(5x+2) 25x^2=(5x)^2 4=2^2 20x
b

We will now rewrite the middle term using the square roots of the perfect squares of the first and last terms.

Polynomial Factored Polynomial First Term Last Term Middle Term
4x^2+12x+9 (2x+3)(2x+3) 4x^2=( 2x)^2 9= 3^2 12x=2( 2x)( 3)
9x^2-24x+16 (3x-4)(3x-4) 9x^2=( 3x)^2 16= 4^2 - 24 = - 2 ( 3x)( 4)
4x^2-20x+25 (2x-5)(2x-5) 4x^2=( 2x)^2 25= 5^2 - 20 = - 2 ( 2x)( 5)
16x^2+24x+9 (4x+3)(4x+3) 16x^2=( 4x)^2 9= 3^2 24x = 2 ( 4x)( 3)
25x^2+20x+4 (5x+2)(5x+2) 25x^2=( 5x)^2 4= 2^2 20x= 2 ( 5x)( 2)
c

Considering the examples in the tables, we can write the followings.

c|c a^2+2 a b+ b^2 & a^2-2 a b+ b^2 ⇕ & ⇕ ( a+ b)^2 & ( a- b)^2
d

For a trinomial to be classified as a perfect square trinomial, we need to ask a few questions.

  1. Is the first term a perfect square?
  2. Is the last term a perfect square?
  3. Is the middle term twice the product of square roots of the first and last term?

If the answer to all of them is yes, then it is a perfect square trinomial. The sign of the middle term determines if it is a square of a sum or a difference. a^2± 2 a b+ b^2 = ( a ± b)^2
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