{ Expanding and Factoring Polynomial Expressions Different ways to look at things.
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Expanding and Factoring Polynomial Expressions
Different ways to look at things.
Polynomial Expressions
Expression—Symbols with meaning
Exs: 1 + 23a2 - bsin(π)
Polynomial Expressions
Polynomial Expression—Expression with these symbols:
+-xvariables coefficientsexponents (limited)
Polynomial Expressions
Which is not one?
a. 1 + 2b. 3a2 - bc. sin(π)
Polynomial Expressions
Which is not one?
a. x2 + y2
b. 3a/bc. 0
Polynomial Expressions
Which is not one?
a. 3a + 2b +4cb. 4a-2.5 + bc. 92 + 8m
Binomial Expansion
Binomial—Polynomial with two terms
Exs: 3a + b3x2 + 4
Binomial Expansion
Expansion? Exponentially increasing a Binomial
See the following:(x + 4)2
Binomial Expansion
How do we expand (x + 4)2?
x2 + 42
= x2 + 16?
No!
FOIL Method
How do we expand (x + 4)2?
We use the FOIL method, an acronym for distribution.
FOIL Method
First Outer Inner Last
FOIL Method
(x + 4)2
= (x + 4)(x + 4)F (x + 4)(x + 4) » x2
O (x + 4)(x + 4) » 4xI (x + 4)(x + 4) » 4xL (x + 4)(x + 4) » 16
= x2 + 4x + 4x + 16= x2 + 8x + 16
FOIL Method
Applies to any multiplication of binomials.
Ex: (x + 2)(3x - 7)
Example in Action
FOIL Method
Practice Problems:
1. (x - 11)22. (2x - 4)(2x + 3)3. (a + b)24. (a + b)(c + d)
Factorization
How do you reverse foil?
A process called factoring!
Factorization
When factoring, you are a detective.
Think about the clues.
Factorization
x2 + 7x + 10
What are the clues?
Recall FOIL and reverse it.
Factorization
x2 + 7x + 10
F: the first term is the product of the two first terms.
What gives us x2?
Factorization
x2 + 7x + 10
F: x and xL: the last term is the product of two terms.
What gives us 10?
Factorization
x2 + 7x + 10
F: x and xL: 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?
Factorization
x2 + 7x + 10
F: x and xL: 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).
Factorization
Use the FOIL method to check your answer: (x + 5)(x + 2)
The result is x2 + 7x + 10 just as it should be.
Factorization
Another example question:
x2-2x-24
Factorization
x2 - 2x - 24
Use the clues:F: What produces x2?
x and x
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
Factorization
x2 - 2x - 24
Use the clues:OI: What makes -2?
-6 and 4
Factorization
x2 - 2x - 24
Putting the clues together makes the answer:
(x - 6)(x + 4)
Factorization
Why?
Why factor? When will you need this?
Solutions!
Why Factor?
• It’s fun to be a detective.• It makes things look pretty.• It gives solutions!
Solutions!
How does it give solutions?
x2 - 2x – 24 = 0(x – 6)(x + 4) = 0x = 6, x = -4
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