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November 23rd









November 23rd

Greatest Common Factor

Textbook Highlight
1. The helpful hint on page 950 is very important. You should always check factoring by multiplying.

Warm Up:
Multiply:
a) 5(x2 + 2x + 3)
b) 5x(x2 + 2x + 3)
c) 5x2(x2 + 2x + 3)

Factoring a Polynomial
• To write a polynomial as a product of factors . Factoring breaks a polynomial down into smaller parts (factors).
• Factoring is the process of deconstructing the polynomial.

The first step in factoring a polynomial is always to factor out the Greatest Common Factor, or GCF.

To find the GCF of a set of numbers:
1. Write each number as the product of primes.
2. Find the smallest power of each common factor .
3. The GCF is the product of all numbers found in step 2.

Note: We will eventually learn to find GCF’s of numbers by looking at the numbers and finding the greatest (largest) number that divides into each number .

Example: Find the GCF.
a) 6 and 8
b) 45 and 75
c)14,24, and 60
d) 32 and 33

We can translate finding the GCF of a set of numbers to finding the GCF of a set of variables .

To find the GCF of a set of variables:
1. Find the smallest power of each common factor.
2. The GCF is the product of all factors found in step 1.

Example: Find the GCF.
a) y6, y5, and y8
b) x, x2, and x3
c) x3y4, x2y3, x5y

To find the GCF of a set of monomials :
1. Find the GCF of the coefficients.
2. Find the GCF of the variables.
3. The product of steps 2 and 3 is the GCF of the monomials.

Example: Find the GCF.
a) 6x2, 20x4, - 8x3
b) 2x, 24
c) 20a2b3, 15ab4

To factor out the GCF of a polynomial:
1. Find the GCF of the polynomial.
2. Divide each term of the polynomial by the GCF.
3. Write the polynomial as the product of the GCF and the results of step 2.
4. Check your result by multiplyin the polynomial back out (at least mentally ).

Example (MML ref 13.1: 1, 2, 3): Factor each polynomial by factoring out the GCF.
a) 5x2 + 10x + 15
b) 5x3 + 10x2 + 15x
c) 5x4 + 10x3 + 15x2
d) 10xy + 5y
e) x3 - x2
f) 4x2 - 6x
g) 4xy + 8x2y
h) 25x2 - 30y2
i) - 2x3y - 6x2y - 10xy

Question: Should we have factored out -2 in problem i?
Rule: In general is the term with the highest degree (the leading term) is negative, we will factor out a negative GCF. Otherwise, we’ll factor out a positive GCF .
If the leading term is negative and has a coefficient of -1, we will factor out a -1.

Example: Factor out the GCF.
a) -3x + 6y
b) -4x2 - 8x + 4
c) -x - 5

Example (MML ref 12.1: 4, 5):
Questions:
What is the GCF of 3(x - 1) + y(x - 1)?
Can we factor 3(x - 1) + y(x - 1) by factoring out the GCF?
Can we factor x(x - 4) - 3(x - 4) by factoring out the GCF?

Factoring by Grouping
Is a technique used to factor some special polynomials with four terms
Is a technique used in section 13.4 to factor certain quadratic (degree 2) polynomials

To factor by grouping:
1. Find the GCF of the first two terms and factor it out.
2. Find the GCF of the second two terms and factor it out, making sure that what is left matches what was left in step 1. In this step, you may have to factor a “1” or “-1” as a place holder.
3. “What was left” becomes the common factor and can now be factored. Factor that out. This becomes the first final factor.
4. The second final factor is the combination of what remains .

Example:
ax + ay + bx + by

Example (MML ref 12.1: 6, 7, 8, 9, 10): Factor by grouping.
a) 2x + 2y + x2 + xy
b) 2y + 8 + xy + 4x
c) 8y2 - 12y + 10y - 15
d) 15a6 - 25a3 - 6a3 + 10
e) 8x2 - 4x + 2x -
 f) 3x2 + 6x - x - 2

Multiple Choice Practice Questions.

1) Factor completely: 36a2b2 - 9ab
A. 9ab(4ab)
B. 9ab(4ab - 1)
C. 9ab(4a2b2 - 1)
D. 3ab(12ab - 3)

2) Factor completely: 6x2 - 6xy - 5x + 5y
A. (6x - 5)(x + y)
B. (6x - y)(x + 5)
C. (6x - 5)(x - y)
D. (6x - y)(x - 5)

Challenge
Factor:
12x2y - 42x2 - 4y + 14 Hint: Remember to factor out a GCF first!

Summary

Concept What Do I Need to Know?
Finding the GCF of a set of numbers The GCF is the largest number that divides evenly into every number in the set.
Finding the GCF of a set of variables The GCF is the product of the smallest powers of the common factors.
Factoring out the GCF of a polynomial To factor out the GCF, find the GCF and rewrite the polynomial as the product of two ex pressions : the GCF and the polynomial with each term having the GCF divided out.
Factoring by grouping To factor a four term polynomial by grouping, factor a common factor out of each pair of terms. That should leave a binomial in common . Factor out that binomial and combine what’s left.


Be sure to work My Math Lab Homework section 13.1.

Suggested Optional Reinforcement Activities
Textbook: Work the concept extensions at the top of page 953.
Answers:
89) b
90) factored
91) factored
92) not factored
93) not factored
Tracked Tutorial problems:
13.1: 59, 61, 62, 65, 67, 69, 71, 73

For next time: Read pages 955 - 958 in your textbook.

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