This document discusses expanding and factoring polynomial expressions. It introduces polynomials and the binomial expansion method of FOIL (First, Outer, Inner, Last). FOIL is used to expand expressions like (x + 4)2 into x2 + 8x + 16. The document also covers factoring polynomials by reversing the FOIL process. Factoring allows one to find the solutions to equations like x2 - 2x - 24 = 0, with solutions x = 6 and x = -4.

1. Expanding and
Factoring Polynomial
Expressions
{
Different ways to look at things.

2. Polynomial Expressions
Expression—Symbols
with meaning
Exs: 1 + 2
3a2 - b
sin(π)

3. Polynomial Expressions
Polynomial Expression—
Expression with these symbols:
+
-
x
variables
coefficients
exponents (limited)

4. Polynomial Expressions
Which is not one?
a. 1 + 2
b. 3a2 - b
c. sin(π)

5. Polynomial Expressions
Which is not one?
a. x2 + y2
b. 3a/b
c. 0

6. Polynomial Expressions
Which is not one?
a. 3a + 2b +4c
b. 4a-2.5 + b
c. 92 + 8m

7. Binomial Expansion
Binomial—Polynomial with two
terms
Exs: 3a + b
3x2 + 4

8. Binomial Expansion
Expansion? Exponentially
increasing a Binomial
See the following:
(x + 4)2

9. Binomial Expansion
How do we expand (x + 4)2?
x2 + 42
= x2 + 16?
No!

10. FOIL Method
How do we expand (x + 4)2?
We use the FOIL method, an
acronym for distribution.

11. FOIL Method
First Outer Inner Last

12. FOIL Method
(x + 4)2
= (x + 4)(x + 4)
F (x + 4)(x + 4) » x2
O (x + 4)(x + 4) » 4x
I (x + 4)(x + 4) » 4x
L (x + 4)(x + 4) » 16
= x2 + 4x + 4x + 16
= x2 + 8x + 16

13. FOIL Method
Applies to any multiplication of
binomials.
Ex: (x + 2)(3x - 7)
Example in Action

14. FOIL Method
Practice Problems:
1. (x - 11)2
2. (2x - 4)(2x + 3)
3. (a + b)2
4. (a + b)(c + d)

15. Factorization
How do you reverse foil?
A process called factoring!

16. Factorization
When factoring, you are a
detective.
Think about the clues.

17. Factorization
x2 + 7x + 10
What are the clues?
Recall FOIL and reverse it.

18. Factorization
x2 + 7x + 10
F: the first term is the product of
the two first terms.
What gives us x2?

19. Factorization
x2 + 7x + 10
F: x and x
L: the last term is the product of
two terms.
What gives us 10?

20. Factorization
x2 + 7x + 10
F: x and x
L: 10 and 1, 5 and 2, -10 and -1,
OR -5 and -2.
OI: the middle term is the sum of
the outer and inner products.
What gives us 7?

21. Factorization
x2 + 7x + 10
F: x and x
L: 10 and 1, 5 and 2, -10 and -1,
OR -5 and -2.
OI: 5 and 2
So, our answer is (x + 5)(x +2).

22. Factorization
Use the FOIL method to check
your answer: (x + 5)(x + 2)
The result is x2 + 7x + 10 just as
it should be.

23. Factorization
Another example question:
x2-2x-24

24. Factorization
x2 - 2x - 24
Use the clues:
F: What produces x2?
x and x

25. Factorization
x2 - 2x - 24
Use the clues:
L: What produces -24?
-24 and 1, -1 and 24, -12 and 2, -2
and 12, -8 and 3, -3 and 8, -6 and
4, OR -4 and 6

26. Factorization
x2 - 2x - 24
Use the clues:
OI: What makes -2?
-6 and 4

27. Factorization
x2 - 2x - 24
Putting the clues together makes
the answer:
(x - 6)(x + 4)

28. Factorization
Why?
Why factor? When will you need
this?

29. Solutions!
Why Factor?
• It’s fun to be a detective.
• It makes things look pretty.
• It gives solutions!

30. Solutions!
How does it give solutions?
x2 - 2x – 24 = 0
(x – 6)(x + 4) = 0
x = 6, x = -4