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Summer 2016 - Session 2
Math 1300
FUNDAMENTALS OF MATH
Section #16535
Monday - Friday, 10am-12pm
Instructor: Dr. Angelynn Alvarez
[email protected]
Section 4.1 – Greatest Common
Factors & Factoring by Grouping
Section 4.1 – GCFs & Factoring by Grouping
- Recall: In section 1.3, we discussed that the greatest common
divisor (GCF) of two integers was the biggest number which divides
both numbers.
- We also said that the words “factors” and “divisors” are synonyms.
- We can also do the same with polynomials: We can find the
greatest common factor of two polynomials --- which is the largest
degree polynomial that divides both polynomials.
Finding the greatest common factor of polynomials:
(1)
(2)
(3)
Find the GCF of the coefficients (like in section 1.3).
Multiply the GCF of the coefficients by the variables that
appear in BOTH terms.
To determine the exponents: The smallest exponent of each
variable is what appears in the GCF.
Example: Find the GCF of 9𝑥 !" and 18𝑥 ! .
Example: Find the GCF of 11𝑥 !" and 22𝑥 !" .
Example: Find the GCF of 15𝑥 ! and 15.
Example: Find the GCF of 21𝑥 ! and 21.
Example: Find the GCF of 7𝑥 ! 𝑦 !" and 49𝑥 !" 𝑦 ! .
Example: Find the GCF of 11𝑥 ! 𝑦 ! and 55𝑥 ! 𝑦 ! .
Example: Find the GCF of 2𝑥 ! 𝑦 ! and 12𝑥 !" 𝑦 ! .
Note: If the coefficients have nothing in common, the GCF of the
coefficients is 1.
Example: Find the GCF of 20𝑥 ! 𝑦 !" and 11𝑥 ! 𝑦 ! .
Example: Find the GCF of 11𝑥 ! 𝑦 !" and 9𝑥 !" 𝑦 !! .
Example: Find the GCF of 64𝑥 ! 𝑦 ! 𝑧 ! and 8𝑥𝑦 ! 𝑧 ! .
Example: Find the GCF of 48𝑥 ! 𝑦𝑧 ! and 12𝑥𝑦𝑧 ! .
Example: Find the GCF of 96𝑥 ! + 72𝑥 ! + 12𝑥 ! .
*Hint: Ignore [for now] the +/−, and find the GCF of all 3 terms.
Example: Find the GCF of 28𝑥 ! − 42𝑥 ! + 7𝑥 ! .
*Hint: Ignore [for now] any +/−, and find the GCF of all 3 terms.
Factoring Polynomials
- Given any polynomial, we want to write the polynomial as a product
of the GCF of the terms and another polynomial:
(polynomial)=(GCF)×(another polynomial)
- This process is called factoring.
To factor a polynomial:
(1)
Ignore all addition and subtraction signs [for now] and
find the GCF of all of the terms.
(2)
Multiply the GCF in step (1) by a polynomial that you
need to get to back the original polynomial that you
started with.
Example: Factor the following polynomials completely.
12𝑥 ! + 12
9𝑥 ! + 9
12𝑥 ! 𝑦 − 36𝑥 ! 𝑦 !
15𝑥𝑦 ! − 20𝑥 ! 𝑦 !
63𝑥 ! 𝑦 − 56𝑥 ! 𝑦 !
Factoring by Grouping
- If a polynomial has 4 or more terms, it helps to group terms
together and factor out the common terms in each group.
- In this class, we will deal with terms with just 4 terms.
- We call this process “Factoring by Grouping”.
How to factor by grouping:
(1)
(2)
(3)
Group the first 2 terms together and the last 2 terms
together. Then factor each group.
*Don’t forget the any “−“!
Identify the term that is common to both groups.
Multiply the common term in step (2) by the sum of the
uncommon terms.
Example: Factor the following polynomials completely.
5 𝑥 − 𝑦 − 𝑡(𝑥 − 𝑦)
9 𝑥 − 𝑦 − 𝑡(𝑥 − 𝑦)
𝑥 𝑥 + 11 − (𝑥 + 11)
𝑥 𝑥 + 9 − (𝑥 + 9)
𝑐𝑦 + 𝑏𝑦 + 𝑐𝑥 + 𝑏𝑥
𝑚𝑏 + 𝑦𝑏 + 𝑚𝑥 + 𝑦𝑥
𝑦𝑚 + 𝑐𝑚 + 𝑦𝑎 + 𝑐𝑎
𝑥 ! − 3𝑥 ! + 6𝑥 − 18
𝑥 ! − 7𝑥 ! + 6𝑥 − 42
𝑥 ! − 5𝑥 ! + 8𝑥 − 40
*Tip for the following examples: Factoring out a “−“ creates a “+”.
That is:
−𝑥𝑦 − 𝑥𝑧 = −𝑥(𝑦 + 𝑧)
Example: Factor the following polynomials completely.
−3𝑥 + 3𝑦 − 𝑥𝑦 − 𝑦 !
5𝑥 + 5𝑦 − 𝑥𝑦 − 𝑦 !
7𝑥 + 7𝑦 − 𝑥𝑦 − 𝑦 !