1. 6.8 Solving Equations by Factoring

2. The Principal of Zero products
For any rational numbers a and b, if ab = 0, then a = 0 or b = 0, and if a = 0 and b = 0, then ab = 0.

3. Example #1 SOLVE (5x + 1)(x – 7) = 0 5x + 1 = 0 x – 7 = 0
NOTE: ALWAYS make sure your answers make you equation true!!! You sub in one value in at a time to check that value and then move on to the other values you got as answers!

4. Example #2
SOLVE x(2x – 9) = 0 x = 0 2x – 9 = 0 2x = 9 x = 9 2

5. You can use the following steps to solve equations using the property of zero Products
Get zero on one side of the equation using the addition property.
Factor the expression on the other side of the equation.
Set each factor equal to zero.
Solve each equation.

6. Example #3 𝑺𝒐𝒍𝒗𝒆 𝒙 𝟐 −𝟖𝒙=−𝟏𝟔 𝑥 2 −8𝑥+16=0 (x – 4)(x – 4) = 0 x – 4 = 0
𝑺𝒐𝒍𝒗𝒆 𝒙 𝟐 −𝟖𝒙=−𝟏𝟔
𝑥 2 −8𝑥+16=0
(x – 4)(x – 4) = 0
x – 4 = 0
x = 4
NOTE: (x – 4)(x – 4) = (𝑥 −4) 2 so you only have one solution
(They would both give you the same answer!!)

7. EXAMPLE #4 𝒙 𝟐 +𝟓𝒙+𝟔=𝟎 (x + 2)(x + 3) = 0 x + 2 = 0 x + 3 = 0

8. Example #5
𝒙 𝟐 −𝟓𝒙=𝟎 x(x – 5) = 0 x = 0 x – 5 = 0 x = 5

9. Example #6
𝟒 𝒙 𝟐 −𝟐𝟓=𝟎 (2x – 5)(2x + 5) = 0 2x = 5 2x = -5 x = 5 2 x = NOTE: If you are asked for roots of a given polynomial you do the SAME THING(means the same thing).