What is the greatest common factor of $x^6$ and $x^9$?
A. $x^3$
B. $x^6$
C. $x^9$
D. $x^{15}$
Answer (2)
Identify the terms: x 6 and x 9 .
Determine the greatest common factor (GCF) by finding the highest power of x that divides both terms.
Use the rule: GCF of x a and x b is x m i n ( a , b ) .
Calculate the minimum of the exponents: min ( 6 , 9 ) = 6 , so the GCF is x 6 .
Explanation
Understanding the Problem We are asked to find the greatest common factor (GCF) of x 6 and x 9 . The GCF is the largest expression that divides both terms without leaving a remainder.
Finding the Greatest Common Factor To find the GCF of x 6 and x 9 , we need to determine the highest power of x that is a factor of both terms. We can express x 6 and x 9 as follows:
x 6 = x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x
x 9 = x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x
The factors of x 6 are x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 .
The factors of x 9 are x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 .
The common factors are x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 . The greatest of these is x 6 .
Using the Minimum Exponent Rule Alternatively, we can use the rule that the GCF of x a and x b is x m i n ( a , b ) , where min ( a , b ) is the minimum of a and b . In this case, a = 6 and b = 9 , so we need to find the minimum of 6 and 9, which is 6. Therefore, the GCF of x 6 and x 9 is x 6 .
Final Answer The greatest common factor of x 6 and x 9 is x 6 .
Examples
Understanding the greatest common factor is useful in simplifying algebraic expressions and solving equations. For example, if you have an expression like x 3 x 9 + x 6 , you can factor out the GCF of x 9 and x 6 , which is x 6 , to simplify the expression: x 3 x 6 ( x 3 + 1 ) = x 3 ( x 3 + 1 ) . This technique is also used in cryptography and data compression to optimize algorithms and reduce storage space.
The greatest common factor of x 6 and x 9 is x 6 , as it is the highest power of x that divides both terms. Thus, the answer is option B: x 6 .
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