Mathematics 506 - Technical & Scientific Option
Lesson 2 – Factoring Polynomials (cont’d)
Methods:
5)
Perfect Square Trinomials
6)
Completing the Square
7)
Roots Method
Perfect Square Trinomials
A trinomial is a perfect square when the middle term is equal to twice the product of the square roots of
the end terms.
𝑎2 + 2𝑎𝑏 + 𝑏 2 = (𝑎 + 𝑏)2
Ex.
OR
9𝑥 2 + 12𝑥 + 4
𝑎2 = 9𝑥 2 ⟹ 𝑎 = 3𝑥
𝑏2 = 4
⟹ 𝑏=2
⟹ 2𝑎𝑏 = 2 3𝑥 2 = 12𝑥
∴ 9𝑥 2 + 12𝑥 + 4 = (3𝑥 + 2)2
3𝑥 + 2 3𝑥 + 2
Verify:
= 9𝑥 2 + 6𝑥 + 6𝑥 + 4
= 9𝑥 2 + 12𝑥 + 4
Ex.
9𝑥 2 − 12𝑥 + 4
𝑎2 = 9𝑥 2 ⟹ 𝑎 = 3𝑥
𝑏2 = 4
⟹ 𝑏=2
⟹ 2𝑎𝑏 = 2 3𝑥 2 = 12𝑥
∴ 9𝑥 2 − 12𝑥 + 4 = (3𝑥 − 2)2
𝑎2 − 2𝑎𝑏 + 𝑏 2 = (𝑎 − 𝑏)2
Mathematics 506 - Technical & Scientific Option
Completing the Square
If the trinomial is not a perfect square we can complete the square by adding a constant.
NB:
This method only works when the coefficient of 𝑥 2 is 1. If it is not, the expression must be
factored (or the equation must be divided by the coefficient) before we can complete the
square.
Ex.
2𝑥 2 − 12𝑥 + 8 = 2(𝑥 2 − 6𝑥 + 4)
To complete the square 𝑥 2 + 𝑏𝑥, add the square of half the coefficient of 𝑥 (ie.
𝑏 2
)
2
to create a perfect
square. We must also subtract this same constant from the expression.
𝑥 2 + 𝑏𝑥 +
Ex.
𝑏 2
2
−
𝑏 2
2
𝑏
+ 𝑐 = (𝑥 + 2 )2 −
𝑏 2
2
+ 𝑐5
2𝑥 2 − 12𝑥 + 8
= 2(𝑥 2 − 6𝑥 + 4)
Step 1: Make the coefficient of 𝑥 2 =1.
Step 2: Find
𝑏 2
2
=
−6 2
2
= 2[(𝑥 2 − 6𝑥 + 9) − 9 + 4]
Step 3: Add (and subtract ) this constant.
= 2[(𝑥 − 3)2 − 5]
Step 4: Factor the perfect square.
= 2(𝑥 − 3)2 − 10
Step 5: Simplify
This method is used mainly when solving equations.
Ex.
=9
2𝑥 2 − 12𝑥 + 8 = 0
2(𝑥 − 3)2 − 10 = 0
Factor as explained above
2(𝑥 − 3)2 = 10
Solve for 𝑥
(𝑥 − 3)2 = 5
𝑥−3=± 5
Case 1: 𝑥 − 3 = 5
𝑥 =3+ 5
Case 2:
𝑥−3=− 5
𝑥 =3− 5
Mathematics 506 - Technical & Scientific Option
Roots Method
Any second degree trinomial (𝑎𝑥 2 + 𝑏𝑥 + 𝑐) can be factored if and only if the discriminant  is positive
or zero, where
= 𝑏 2 − 4𝑎𝑐
If  ≥ 0 then 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 𝑎 𝑥 − 𝑥1 𝑥 − 𝑥2
and
𝑥1 =
−𝑏+ △
2𝑎
=
−𝑏+ 𝑏 2 −4𝑎𝑐
2𝑎
This method is also mainly used to solve equations.
𝑥2 =
−𝑏− △
2𝑎
=
−𝑏− 𝑏 2 −4𝑎𝑐
2𝑎