Download 4.2 factors and simplest form

4.2 FACTORS AND SIMPLEST FORM
As we said before, when we SIMPLIFY a fraction, we wish
to write its equivalent using the smallest numbers possible.
This is called writing the fraction in LOWEST TERMS.
The numerator and denominator should have no common
factors (that can be divided out!) other than 1.
What are some “shortcuts” to know if a number is
divisible by:
2
3
4
5
6
8
9
10
Ex:
Fraction Lowest Terms?
Common
Factor(s)?
Simplified to
Lowest Terms:
5
7
14
21
_ 8
16
Notice that we want to divide out the LARGEST
COMMON FACTOR possible. If we don’t at first, we just
KEEP DIVIDING until there are no more common factors
(except 1).
Ex: Reduce to lowest terms:
_ 5400
18000
Again, notice that the SIGN always remains the SAME for
equivalent fractions.
We can also use PRIME FACTORIZATION to divide out
all possible common prime factors at once. We would then
multiply together the final result knowing that there are no
more common factors. First we need to know what prime
factorization is!!
PRIME NUMBERS are whole numbers that have exactly
TWO DISTINCT factors, ITSELF and 1. Ex: 2, 3, 5, 7,
_____, ______, _____. (For those interested, see me for a
way to generate a list.)
COMPOSITE NUMBERS are numbers with at least one
other factor BESIDE itself and 1. Ex: ____, ____, ____
Note that the number ONE is _____ a prime factor. Why?
The number ZERO is _______ a prime number. Why?
As a matter of fact, zero and one are NEITHER PRIME
NOR COMPOSITE!!
Now we can define PRIME FACTORIZATION of a
number, it is factorization where every factor is a PRIME
NUMBER.
Ex: 8 = 2 4 is NOT the prime factorization of 8.
8 = 2 3 is NOT any factorization of 8.
Write 8 = 222 (or 23 ) to write its PRIME factorization.
Lets see how this helps us write fractions in lowest terms,
then we will explore how to easily “break it down”.
Ex: Write in lowest terms
180
220
= 22335
2  2  5 11
= 33
11
= 9
11
To find the prime factorization of a number we can either
use DIVISION by prime numbers, or we can use
FACTORING TREES. When we get to the end, every
factor should be prime. Lets try a couple:
144
3850
To use prime factorization to write fractions in lowest
terms (simplest form):
1) Write the prime factorization of both the Numerator
and the Denominator
2) Divide out common factors and replace them with
the number ONE (to hold the place in
multiplication, recall 11 = 1, not 2!)
***Recall x3 means x  x  x
3) Multiply the remaining factors in the Numerator
and Denominator.
Ex: Reduce by using prime factorization:
a)
18
36
b)
36
18
c)
96

288
d)
6
3x
5
e)
8b c
20abc
Next we will look at EQUIVALENT FRACTIONS. These
are fractions which represent the same number (point) on
the number line. (Insert from old 4.1.)
List equivalent fractions for ½. What do you notice?
Two fractions are equivalent if:
Do p. 236~ # 52, 62, 72, 94.