Download Fundamentals of Mathematics - §4.1

Fundamentals of Mathematics
§4.1 - Greatest Common Factor and
Factoring by Grouping
Ricky Ng
November 8, 2013
Ricky Ng
Fundamentals of Mathematics
Announcements
1
HW 16 is due on Sunday.
2
Online quiz is due today.
Ricky Ng
Fundamentals of Mathematics
Why Chapter 4?
We have seen how to multiply polynomials such as
(2x + 3)(x
2) = 2x2
x
6.
How about going backward? That is, if we just start with
2x2
x
6,
how do we retrieve the product (2x + 3)(x 2)?
This process is known as factoring. We will spend most of the
chapter learning how to do that.
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§4.1 - GCF and Grouping
GCF of Monomials
Just as multiplication, we will start with monomials.
Definition (Greatest Common Factor)
The greatest common factor (GCF) of two monomials is
the greatest factor that divides both monomials.
This is analogous to GCD of numbers.
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§4.1 - GCF and Grouping
How to find GCF?
To find out the GCF, it basically reduces to finding
GCD of the coefficients.
Common terms in highest degree.
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§4.1 - GCF and Grouping
Recap on GCD
Recall that the GCD of two natural numbers is the greatest
number that divides both numbers. As a quick example, find
gcd(24, 36).
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§4.1 - GCF and Grouping
Popper 21: Question 1
Find gcd(48, 80).
a)
b)
c)
d)
e)
4
8
12
16
6
Ricky Ng
§4.1 - GCF and Grouping
Recap on xn
Recall that xn means
x
| ⇥x⇥
{z· · · ⇥ x} .
n times
So x2 divides x3 because
x3 = x · x · x = x2 · x.
Similarly, if k  n are positive integers, then xk divides xn .
(Count how many x’s are multiplied.)
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§4.1 - GCF and Grouping
Examples
Find the GCF of 18x2 , 24x4 , and 12x3 .
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§4.1 - GCF and Grouping
Find the GCF of 25x3 y 4 , 60x4 y 3 , and 15x2 y 4 .
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§4.1 - GCF and Grouping
Popper 21: Question 2
Find the GCF of 16x4 y 3 z 5 , 40x5 y 4 z 4 , and 200x6 y 2 z 5 .
a)
b)
c)
d)
e)
4x4 y 4 z 4
8x4 y 3 z 5
16x5 y 3
8x4 y 2 z 4
4x3 y 3 z 3
Ricky Ng
§4.1 - GCF and Grouping
Ricky Ng
§4.1 - GCF and Grouping
Factoring with GCF
The first step to factor a polynomial is find out the GCF of its
terms. For example, consider the polynomial
p(x) = 4x4 + 6x3 .
The GCF of 4x4 and 6x3 is 2x3 since
4x4 = 2x3 ⇥ 2x
6x3 = 2x3 ⇥ 3.
Hence, using the distributive property of multiplication
4x4 + 6x3 = 2x3 ⇥ 2x + 2x3 ⇥ 3
= 2x3 ⇥ (2x + 3)
= 2x3 (2x + 3)
Ricky Ng
§4.1 - GCF and Grouping
Examples
Factor 4x3
8x2 + 12x by GCF.
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§4.1 - GCF and Grouping
Factor 15xy 2
65x2 y by GCF.
Ricky Ng
§4.1 - GCF and Grouping
Popper 21: Question 3
Factor
4x2 y 5
16x3 y 4 + 24x4 y 2
by GCF.
a) 4xy 2 (y 4 + 4xy + 6x2 )
b) 4x2 y 2 (y 3 4xy 2 + 6x2 )
c) 4x2 y 2 (xy 2 4xy + 6x2 y)
Ricky Ng
§4.1 - GCF and Grouping
Ricky Ng
§4.1 - GCF and Grouping
Factoring by Grouping
Another technique we use is grouping.
Rule (Grouping)
When a polynomial has 4 or more terms, try to rearrange the
terms into 2 or more groups. Then find the GCF of each
group to look for a common factor.
As a quick example, the polynomial x(x + 6) + 4(x + 6) is
already in such a form. Then by observing (x + 6) is a common
factor,
x(x + 6) + 4(x + 6) = (x + 4)(x + 6).
Ricky Ng
§4.1 - GCF and Grouping