Factoring Polynomials to find its zeros

1. Factoring Polynomials
to find zeros
Miss Ramirez
Algebra 1 /9th grade

2. Lecture Objectives and CA Content Standards
• Lecture Objectives
- Learn how to find the GCF
- Factor the greatest common factor of a polynomial
-Factor simple quadratics to find its zeros
• CA Content standards
- Standard A-SSE.1 : Interpret the structure of expressions.
- Standard A-SSE.3a: Factor a quadratic expression to reveal the zeros
of the function.

3. Educational Hook
• In this unit we have learned the parts of a polynomial
expression and now we are going to take it a step further and
learn how to factor polynomials to find its zeros.

4. Know the parts of a polynomial expression
5𝑥2 + 2𝑥 − 7
Degree: 2
Leading Coefficient: 5
Leading Term: 5𝑥2
Constant Term: -7

5. Greatest Common Factor
• The Greatest Common Factor: The largest polynomial that divides evenly into
the polynomials
• When factoring a polynomial always check for a GCF first.
Example 1: 6𝑥2
+ 18𝑥
6= 1, 2, 3, 6
18=1, 2, 3, 6, 18
• The greatest common factor is 6
• The greatest common variable is 𝑥
• The GCF of the polynomial is 6𝑥

6. Greatest Common Factor cont.
• Once you find the GCF you then rewrite the polynomial with the
remaining factors
6𝑥(𝑥 + 3)
• Now you can find the zeros of this polynomial by setting each term
equal to zero.
6𝑥 = 0 𝑥 + 3 = 0
𝑥 = 0 𝑥 = −3
• The zeros of 6𝑥2
+ 18𝑥 are 𝑥 = 0 and 𝑥 = −3

7. Progress Monitoring Question 2:
• Now you try… Find the greatest common factor for the polynomials
Problem 1: 2𝑥2 − 8𝑥
Problem 2: 5𝑥2 − 5𝑥 − 10

8. Factoring using Difference of two squares
Use the difference of two squares formula
when an expression is made up of a
squared term subtracted from another
squared term.
𝒂𝟐
− 𝒃𝟐
= (𝒂 + 𝒃)(𝒂 − 𝒃)
Example 1: 9𝑥2 − 4
(3𝑥)2−(2)2
(3𝑥 + 2)(3𝑥 − 2)
Example 2: 16𝑥2
− 25
(4𝑥)2
−(5)2
(4𝑥 − 5)(4𝑥 + 5)

9. Progress Monitoring Question 2
• Now you try…. Factor the polynomial expression using the difference of
square formula.
𝑡2
− 25

10. Factoring Simple Quadratics
𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0
• Find two numbers that multiply to give ac and adds to give b
Example: 2𝑥2 + 7𝑥 + 3
ac=2 × 3 = 6
b=7
We want to find two numbers that multiply together to make 6 and add up
to 7.

11. Factoring Simple Quadratics cont.
Example: 2𝑥2 + 7𝑥 + 3
• Write down factors of 6 that add up to 7
Factors of 6: 1, 2, 3, 6
6+1=7
• Rewrite the middle term (7x) as 6x+1x
2𝑥2 + 6𝑥 + 1𝑥 + 3

12. Factoring Simple Quadratics cont.
• Factor the first two terms and the last two terms separately
2𝑥2 + 6𝑥 𝑎𝑛𝑑 1𝑥 + 3
2𝑥 𝑥 + 3 + 1 𝑥 + 3
• Since (x+3) is common in both terms than it can be factor like this.
(2𝑥 + 1)(𝑥 + 3)
• Your final answer is (2𝑥 + 1)(𝑥 + 3)

13. Finding zeros
• You have just learned how to factor now you will set up each factor
equal to zero to find its zeros.

14. Finding the zeros of (2x+1) (x+3)
• Set each factor equal to zero to find its zeros
2𝑥 + 1 = 0 𝑥 + 3 = 0
−1 = −1 −3 = −3
2𝑥 = −1 𝑥 = −3
𝑥 = −
1
2
𝑥 = −3
The zeros of 2𝑥2 + 7𝑥 + 3 are 𝑥 = −
1
2
and 𝑥 = −3

15. From what you have learned from this lecture
Now you try…
•Factor the polynomial and find its zeros
𝑥2
+ 10𝑥 + 9

16. A video for additional help