Which of the following is the greatest common factor of the terms $48 s^5 t^3 u^2$ and $40 s^3 t u^4$?
A) $8 s^2 t^2 u$
B) $8 s^3 t u^2$
C) $6 s^2 t^2 u$
D) $8 s^3 t u^2$
Answer (2)
Find the greatest common factor (GCF) of the coefficients 48 and 40, which is 8.
Identify the lowest powers of each variable present in both terms: s 3 , t 1 , and u 2 .
Multiply the GCF of the coefficients by the lowest powers of the variables: 8 s 3 t u 2 .
The greatest common factor of the given terms is 8 s 3 t u 2 .
Explanation
Problem Analysis We are asked to find the greatest common factor (GCF) of the terms 48 s 5 t 3 u 2 and 40 s 3 t u 4 . The GCF is the largest expression that divides both terms without leaving a remainder. To find the GCF, we need to find the GCF of the coefficients and the lowest power of each variable present in both terms.
GCF of Coefficients First, let's find the GCF of the coefficients 48 and 40. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. The greatest common factor of 48 and 40 is 8.
Lowest Powers of Variables Now, let's find the lowest power of each variable present in both terms:
For s , the powers are s 5 and s 3 . The lowest power is s 3 .
For t , the powers are t 3 and t 1 . The lowest power is t 1 (or simply t ).
For u , the powers are u 2 and u 4 . The lowest power is u 2 .
Combining the Factors Now, we multiply the GCF of the coefficients by the lowest powers of the variables to find the GCF of the terms: 8 ⋅ s 3 ⋅ t ⋅ u 2 = 8 s 3 t u 2
Final Answer Therefore, the greatest common factor of 48 s 5 t 3 u 2 and 40 s 3 t u 4 is 8 s 3 t u 2 .
Examples
Understanding the greatest common factor (GCF) is very useful when simplifying fractions in algebra. For example, if you have an expression like 40 s 3 t u 4 48 s 5 t 3 u 2 , finding the GCF ( 8 s 3 t u 2 ) allows you to simplify the expression to 5 u 2 6 s 2 t 2 . This skill is also helpful in real-world scenarios, such as determining the largest size of square tiles that can be used to cover a rectangular floor without cutting any tiles, where the dimensions of the floor are related to the terms in the problem.
The greatest common factor of the terms 48 s 5 t 3 u 2 and 40 s 3 t u 4 is 8 s 3 t u 2 , which is option D. To find it, we calculated the GCF of the coefficients and identified the lowest powers of the variables. This leads us to the final result of 8 s 3 t u 2 .
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