Description:
Factor questions test your knowledge of both numeric and algebraic factors.
Approach:
On easy questions, build a factor tree to determine all of a number's prime factors. On
harder algebraic questions, knowing a few key algebraic factors makes all
the difference.
1) Factor Tree
For example:
-
56
-
2 * 28
-
2 * 2 * 14
-
2 * 2 * 2 * 7
Thus 2 and 7 are the prime factors of 56.
2) Least Common Multiple (LCM)
The least common multiple is the smallest number into which two numbers divide evenly.
For example:
What is the least common multiple of 56 and 36?
First complete a factor tree for both numbers.
-
56
-
2 * 28
-
2 * 2 * 14
-
2 * 2 * 2 * 7
-
36
-
2 * 18
-
2 * 2 * 9
-
2 * 2 * 3 * 3
In order to determine the LCM, you must include the prime factors from
both numbers. When both numbers have the same prime factor, include the
greater number of factors. In this case 56 has three 2s and 36 has two
2s, so the LCM must have three 2s.
Least common multiple = 2 * 2 * 2 * 3 * 3 * 7 = 504
2-1. Practice:
What is the least common multiple of 72 and 84?
3) Greatest Common Factor (GCF)
The Greatest Common Factor is the largest number by which both numbers can
be divided evenly.
For example:
What is the greatest common factor of 56 and 36?
Remember that you need to complete a factor tree for both numbers.
-
56
-
2 * 28
-
2 * 2 * 14
-
2 * 2 * 2 * 7
-
36
-
2 * 18
-
2 * 2 * 9
-
2 * 2 * 3 * 3
In order to determine the GCF, you must include the shared prime factors
from both numbers. When both numbers have the same prime factor, you include
the smaller number of factors. In this case 56 has three 2s and 36 has
two 2s, so the GCF must only have two 2s.
Greatest Common Factor = 2 * 2 = 4
3-1. Practice:
What is the greatest common factor of 700 and 140?
4) Algebraic Factors
The algebraic factors in the chart below must be memorized. Problems
involving these factors become nearly impossible if you try to solve
for x and y explicitly.
|
Quadratic Formula
|
Algebraic Factors
|
|
x2
+ 2xy + y2
|
(x + y) * (x + y) = (x + y)2
|
|
x2
- 2xy + y2
|
(x - y) * (x - y) = (x - y)2
|
|
(x2
- y2)
|
(x + y) * (x - y)
|
4-1. Practice:
If xy = 12 and x2
+ y2
= 25, what is the
value of (x + y)2
?
5) Solutions and Factors
You can generate algebraic factors for a quadratic equation if you are given
solutions to the quadratic. If a quadratic equation has solutions 5 an 8, then
its factors are (x - 5) and (x - 8)
5-1. Practice:
What are the factors of an equation that has solutions a
and 2b?
