a Complete the table by factoring each polynomial.
B
b Write the middle terms using the square roots of the perfect squares of the first and last terms.
C
c Consider the table in Part B. Do you see a pattern?
D
d Describe the common characteristics of the polynomials in the table.
A
aTable:
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
bTable:
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
c a^2± 2ab+b^2=(a± b)^2
D
d See solution.
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.
Is the first term a perfect square?
Is the last term a perfect square?
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
