Download Factor Perfect Square Trinomials

Factor Perfect Square Trinomials
When factoring a trinomial, we can always use the techniques that we
have previously discussed. However, recognizing a perfect square
trinomial can be quite useful. A perfect square trinomial’s first and last
terms are perfect squares (4, 9, 16, 25, 36,....) and the middle term is
twice the square roots of the first and last terms.
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
It sounds and looks difficult but is actually a shortcut. We just need to
use the square roots of the first and last terms, check the middle term
through multiplication, and use the middle term’s sign.
Example 1: Factor: x2 – 6x + 9
The first and last terms are perfect squares: x2 = x·x and 9 = 3·3
Check the middle term: 2·x·3 = 6x
Use the middle term’s sign: negative
(x – 3)2
To check the answer, multiply: (x – 3)2 = (x – 3)(x – 3)
x(x – 3) – 3(x – 3)
x2 – 3x – 3x + 9
x2 – 6x + 9 Correct
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 2: Factor: 49a2 + 70ab + 25b2
The first and last terms are perfect squares: 49a2 = 7a·7a and 25b2 = 5b·5b
Check the middle term: 2·7a·5b = 70ab
Use the middle term’s sign: positive
(7a + 5b)2
To check the answer, multiply: (7a + 5b)2 = (7a + 5b)(7a + 5b)
7a(7a + 5b) + 5b(7a + 5b)
49a2 + 35ab + 35ab + 25b2
49a2 + 70ab + 25b2 Correct
Example 3: Factor: 4y2 – 60y + 81
The first and last terms are perfect squares: 4y2 = 2y·2y and 81 = 9·9
Check the middle term: 2·2y·9 = 36y ≠ 60y
This is not a perfect square trinomial
Try another method:
Multiply first and last numbers: 4 · 81 = 324
Factors of 324: 1,324; 2,162; 3,108; 4,81; 6,54; 9,36; 12,27; 18,18
Add to get -60. Since the product is positive, both numbers must be negative:
-1+(-324) = -325 No
-2+(-162) = -164 No
-3+(-108) = -111 No
-4+(-81) = -85 No
-6+(-54) = -60 Yes
4y2 – 6y – 54y + 81
Rewrite and expand -60y
2
4y – 6y – 54y + 81
Split the polynomial in half
2y(2y – 3) – 27(2y – 3) Factor the GCF from each side
(2y – 3)(2y – 27)
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 3: Factor: z2 + 12z – 36
The last term is negative, so it is not a perfect square
This is not a perfect square trinomial
Try another method:
Factors of 36: 1,36; 2,18; 3,12; 4,9; 6,6
Add to get 12. Since the product is -36, one number must be negative:
-1+36 = 35 or -36+1 = -35 No
-2+18 = 16 or -18+2 = -16 No
-3+12 = 9 or -12+3 = -9 No
-4+9 = 5 or -9+4 = -5
No
Since no set of factors add to 12, the trinomial cannot be factored.
Prime
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)