What is the greatest common factor of the terms in the expression [tex]$36 a+54 a b-27 b$[/tex]?
Answer (2)
The greatest common factor of the expression 36 a + 54 ab − 27 b is 9, which is derived from the prime factorization of the coefficients and identifying common factors. No common variables were found among all terms. Thus, the GCF consists only of the common numerical factor 9.
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The greatest common factor of the terms in the expression 36 a + 54 ab − 27 b is found by:
Finding the prime factorization of the coefficients: 36 = 2 2 × 3 2 , 54 = 2 × 3 3 , 27 = 3 3 .
Identifying the common prime factors and their lowest powers: The common prime factor is 3, and the lowest power is 3 2 = 9 .
Identifying common variables: There are no variables common to all three terms.
Multiplying the common factors: The GCF is 9.
Therefore, the greatest common factor is 9 .
Explanation
Understanding the Problem We are asked to find the greatest common factor (GCF) of the terms in the expression 36 a + 54 ab − 27 b . The terms are 36 a , 54 ab , and − 27 b .
Prime Factorization of Coefficients First, let's find the prime factorization of the coefficients:
36 = 2 2 × 3 2 54 = 2 × 3 3 27 = 3 3
Identifying Common Prime Factors Now, we identify the common prime factors and their lowest powers among the coefficients. The common prime factor is 3. The lowest power of 3 that appears in all the factorizations is 3 2 = 9 . The prime factor 2 is not common to all three terms.
Identifying Common Variables Next, we look at the variables. The variable 'a' appears in the first two terms ( 36 a and 54 ab ) but not in the third term ( − 27 b ). The variable 'b' appears in the second and third terms ( 54 ab and − 27 b ) but not in the first term ( 36 a ). Therefore, there are no common variables in all three terms.
Finding the GCF Finally, we multiply the common prime factors to find the GCF. In this case, the GCF is just the common numerical factor, which is 9.
Conclusion Therefore, the greatest common factor of the terms 36 a , 54 ab , and − 27 b is 9.
Examples
Understanding the greatest common factor is useful in simplifying expressions and solving problems in various fields. For instance, if you are designing a rectangular garden and want to divide it into equal square plots for planting different types of flowers, finding the GCF of the garden's length and width will help you determine the largest possible size of the square plots. This ensures efficient use of space and simplifies the planting process. Similarly, in manufacturing, determining the GCF of different component sizes can optimize material usage and reduce waste.
