What is the greatest common factor of $x^4$ and $x^5$?
A. $x^4$
B. $x^3$
C. $x^2$
D. $x
Answer (2)
The problem asks for the greatest common factor (GCF) of x 4 and x 5 .
We express x 4 and x 5 in their expanded forms.
We identify the common factors in both expressions, which is x 4 .
Therefore, the greatest common factor of x 4 and x 5 is x 4 .
Explanation
Understanding the Problem We are asked to find the greatest common factor (GCF) of x 4 and x 5 . The GCF is the largest term that divides both expressions without leaving a remainder.
Expanding the Terms Let's express x 4 and x 5 in their expanded forms:
x 4 = x ⋅ x ⋅ x ⋅ x x 5 = x ⋅ x ⋅ x ⋅ x ⋅ x
Identifying Common Factors Now, we identify the common factors in both expressions. Both x 4 and x 5 have four factors of x in common. Therefore, the greatest common factor is x ⋅ x ⋅ x ⋅ x = x 4 .
Final Answer Thus, the greatest common factor of x 4 and x 5 is x 4 .
Examples
Understanding the greatest common factor is useful in simplifying algebraic expressions and solving equations. For example, when you have a fraction like x 4 x 5 , you can simplify it by dividing both the numerator and the denominator by their greatest common factor, which is x 4 . This simplifies the fraction to 1 x = x . This concept is also used in various engineering and physics applications where simplification of complex equations is required.
The greatest common factor of x 4 and x 5 is x 4 . This is because both expressions share the common factor of x to the lowest power, which is 4. Therefore, the correct answer is A. x 4 .
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