Factor and Solve Polynomial Equations-- Notes
Consider the following:
ab=0
Question: If I multiply two things together and the result is zero, what can I say about
those two things?
Answer: The only way to multiply and get zero is to multiply by _______. This is called
"The ______________________ Property"
Let's apply this property to some factors:
(x – 3)(x – 4) = 0
(x + 2)(x + 3) = 0
x(x + 5) = 0
8x(x + 2) = 0
(2x – 3)(x + 1) = 0
5(x – 7)(x + 10)=0
Two types of polynomial equations:
Quadratic Equation
Binomial Equation
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎
𝒂𝟐 − 𝒃𝟐 = 𝟎
Factor
Set each factor
equal to zero
Solve each factor for
the variable
Solutions:
Find the solutions of each polynomial.
1. x2 − 4x − 5 = 0
Roots:
2.
0 = 4x2 + 9x + 5
3. 2𝑥 2 + 6𝑥 = 0
Find the roots of each function.(Set equal to 0 by replacing f(x))
3. f(x) = 3x2 + 2x − 8
4.
f(x) = x2 − 64
5. 𝑓(𝑥) = 10𝑥 2 − 5𝑥
Summarize:
5x2 + 28x − 12 = 0
(5x−2)(x+6) =0
x= {
2
5
, -6}
If you know your solutions/roots, you can find the factors of the polynomial:
Start your parenthesis
(x
)(x
Solutions/Roots
)
Switch the sign of each
root and plug it into your
parenthesis
Factors:
12, 7
-5, -10
3, -1
2
3
,5
−1
2
,4
20, 6
4
-4, 3
If there is a denominator, slide it
in front of your variable