Page 386 - Algebra 1

c. Find the LCM of (60x3 + 24x) and (45x4 + 18x2). SOLUTION
Factor each binomial, if possible.
The GCF of the terms in (60x3 + 24x) is 12x. Factor it. (60x3 +24x)=2·2·3·x(5x2 +2)
The GCF of the terms in (45x4 + 18x2) is 9x2. Factor it.
(45x4 +18x2)=3·3·x·x(5x2 +2)
(5x2 + 2) is a common factor, appearing one time in each binomial. The numbers 2, 3, and the variable x are also factors, appearing at most two times.
LCM=2·2·3·3·x·x(5x2 +2) The LCM is 36x2(5x2 + 2).
Application: Scheduling
A math test is given every 9 days. A history test is given every 14 days. A science test is given every 18 days. How many days into the school year will all three tests be given on the same day?
SOLUTION
Understand The frequency of the tests is a regular pattern. At some point the patterns will overlap and all three tests will be given on the same day.
Plan Math tests are given on days that are multiples of 9. History tests are given on days that are multiples of 14. Science tests are given on days that are multiples of 18. If the LCM of 9, 14, and 18 is found, it will show the day that all three tests will be given.
Solve Write each number as a product of prime numbers.
9=3·3
14 = 2 · 7
18 = 3 · 3 · 2
In the LCM, the factor 2 will appear one time, the factor 3 will appear two times, and the factor 7 will appear one time.
LCM = 2 · 3 · 3 · 7
LCM = 126
So, all three tests will be given 126 days into the school year.
Check List the multiples of each number to 126.
9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126
14: 14, 28, 42, 56, 70, 84, 98, 112, 126 18: 18, 36, 54, 72, 90, 108, 126
Caution
Terms in parentheses are grouped and cannot be separated during factoring. The grouped terms make one factor.
Example
5
Math Reasoning
Predict If a school year is 180 days long, how many times will a student have a math and science test on the same day?
The least multiple that appears in each list is 126. ✓
Lesson 57 371