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.
