Download Notes on Greatest Common Factor - Page I

Notes on Greatest Common Factor - Page I
Name_________________________
MM1A2. Students will simplify and operate with radical expressions, polynomials,
and rational expressions.
f. Factor expressions by greatest common factor, grouping, trial and error, and
special products limited to the formulas below.
To begin, let's review how to divide monomials. In order to divide two monomials, one divides
(or simplifies) the coefficients and subtracts the exponents of like variables.
For example, simplify 8c 7  2c 5 .
First, one divides the coefficients ( 8  2  4 ). Next, one subtracts the exponents of like variables
( c 7  c 5  c 2 ).
2
Put them together, and you get 4c .
--------------------------------------------------------------------------------------------------------------------Other Examples - For all examples, assume the variables do not equal zero.
3d 12
15d 16

2
10d 4
Note in this example that the coefficients (15 and 10) do not divide evenly.
15
3
However, if treated as a fraction ( ), the fraction can simplify to .
10
2
9k 3
9k 4

7
7k
Here - the coefficients (9 and 7) do not divide evenly, nor can they be
simplified. Note how the k in the denominator technically has a "1" for
an exponent. Thus, k 4  k 1 produces k 3 .
One must recall that r 0  1 . In this example, r 5  r 5  r 0 . This, as a
r s  r  1s  s
result, makes the "1" in front of s 3 . Also, note that since s 3 is not
divided by a quantity involving s, it stays the same.
--------------------------------------------------------------------------------------------------------------------For Questions 1-3, simplify completely. Assume all variables do not equal zero.
5 3
5
3
1. 24 x8  4 x 2
3
2.
56 y
7y
3.
5a 20bz 3
3a 4 z 2
--------------------------------------------------------------------------------------------------------------------An integer is any number that does not have digits to the right of the decimal point.
For example, "4" is an integer. " 8 " is an integer. "0" is an integer.
However, "2.71" is not an integer, because of the digits ("71") to the right of the decimal.
1
is not an integer, because when converted to a number, it is 0.5. So, there is a digit (5) to the
2
right of the decimal.
In this unit, we will only deal with integers. Thus, no quantities involving "decimals" (for lack
of a better term) will be used.
The factors of an integer are all of the positive integers that divide evenly into it.
Consider the number 6. Its factors are 1, 2, 3, and 6. Each of these numbers divides evenly
into 6. For example, "2" is one of the factors because it divides into 6 three times. However,
4 would not be a factor of 6, because 4 does not divide evenly into 6.
Consider 12 . Its factors are 1, 2, 3, 4, 6, and 12. Note that we refer to the factors as positive,
even though the original number is negative.
--------------------------------------------------------------------------------------------------------------------For Questions 4-6, list all of the factors for the given number.
4.
15
5.
9
6.
24
--------------------------------------------------------------------------------------------------------------------The greatest common factor, or GCF, of two or more integers is, as one might guess, the
largest factor that they share.
For example, what is the greatest common factor of 10 and 25?
The factors of 10 are 1, 2, 5, and 10.
The factors of 25 are 1, 5, and 25.
The largest number to appear on both lists is "5". So, the GCF is 5 .
--------------------------------------------------------------------------------------------------------------------What is the GCF of 12, 36, and 48?
The factors of 12 are 1, 2, 3, 4, 6, and 12.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
Therefore, the GCF is 12 , because it is the largest factor of all three numbers.
--------------------------------------------------------------------------------------------------------------------A common technique is to find all of the factors of the smallest number. Then, determine the
largest of these factors that is also a factor of the other numbers. This can save a lot of time.
--------------------------------------------------------------------------------------------------------------------For Questions 7-9, find the greatest common factor of the given numbers.
7. 16 and 18
8. 24, 36, and 56
9. 22 and 27
Notes on Greatest Common Factor - Page II
Name_________________________
Finding the greatest common factor of a variable quantity is similar.
For example, what is the GCF of x3 and x 4 ?
Consider that x3  x  x  x , and x 4  x  x  x  x .
Through this expansion, one can clearly see all of the common factors of both quantities. Note
how x  x  x appears in both expansions. Hence, it is the GCF: x  x  x  x .
--------------------------------------------------------------------------------------------------------------------For Questions 10-12, find the GCF of the variable quantities.
3
10. x 2 and x 5
11. z and z10
12. a 3 and b6
--------------------------------------------------------------------------------------------------------------------Let's summarize the findings from 10-12:
The greatest common factor of two or more like variable quantities is the variable raised to the
lowest exponent in all of the quantities.
70
For example, the GCF of c88 , c91 , and c 70 is c , because "70" is the lowest exponent.
--------------------------------------------------------------------------------------------------------------------For Questions 13-15, find the greatest common factor of the quantities below.
13. d 100 and d 56
14. k 7 , k 5 , and k 11
15. p 4 and q 4
--------------------------------------------------------------------------------------------------------------------In order to find the greatest common factor of two or more monomials, simply find the greatest
common factor of the coefficients, and then find the GCF of the variable parts.
For example, what is the GCF of 6x 4 y 7 and 8 y 3 ?
The GCF of 6 and 8 is 2 .
3
The GCF of x 4 y 7 and y 3 is y .
3
Thus, the GCF of the original quantities is 2 y .
--------------------------------------------------------------------------------------------------------------------For Questions 16-18, find the greatest common factor of the quantities below.
16.
5x and 30x 2
17. 10a 5b and 4a 4
18. 27 y 6 , 18y 3 , and 45y 2
Homework on Greatest Common Factor
Name_________________________
For Questions 1-9, simplify completely. You are NOT finding the greatest common factor.
Assume all variable represent nonzero quantities. Refer to Page I of your notes if necessary.
1. 14 x  2 x
2. 64 y  4 y
4
10
5z 2
3.
5z 2
2
4.
24a 8
2
5. b88  b 22
6.
7.
50k 7
10k 7
8. m  m
9.
19c18
3c5
42 p 4 q 5
14q
--------------------------------------------------------------------------------------------------------------------For Questions 10-27, find the greatest common factor of the given quantities. If the greatest
common factor is '1', write PRIME.
10. 13 and 65
11. 72 and 35
12. 96, 48, and 56
13. 9, 36, and 126
14. r and r 3
15. s10 and s8
16. t 12 , t 7 , and t 21
17. u 5 and w30
18. xy 4 and y 5
19. a11b 5 and a10b10
20. c12 d 21e8 , c9 de8 , and c12 d 2 e8
21. 6k and 20k
22. 30m3 and 15m 2
23. 6x3 , 4x 2 , and 2x
24. y 5 , 7 y 4 , and 21y3
25. 18a 2b 2 and 40a 2b 2
26. 12x9 , 5 y 4 , and 7z
27. 22cd and 33d
--------------------------------------------------------------------------------------------------------------------1. 7 x 3
2. 16y8
3. 1
7. 5
8. 1
9. 3 p 4 q 4
8
13. 9
14. r
15. s
19. a10b5
20. c9 de8
21. 2k
25. 2a 2b 2
26. prime
27. 11d
4. 12a8
5. b 66
10. 13
11. prime
19c13
3
12. 8
17. prime
18. y 4
23. 2x
24. y 3
16. t
7
22. 15m 2
6.