To simplify the expression \( 64x^3 - 16x^2 + 8x \), we can start by factoring out the greatest common factor (GCF) from each term.
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Identify the GCF: The GCF of \( 64, -16, \) and \( 8 \) is \( 8 \), and the lowest power of \( x \) present in the terms is \( x \) (from \( 8x \)). Therefore, the GCF is \( 8x \).
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Factor out the GCF:
\[ 64x^3 - 16x^2 + 8x = 8x(8x^2 - 2x + 1) \]
The expression inside the parentheses \( 8x^2 - 2x + 1 \) is a quadratic expression.
Next, we check if \( 8x^2 - 2x + 1 \) can be simplified further, but it does not factor easily into simple integer factors.
Thus, the fully simplified form of the expression, including factoring out the GCF, is:
\[ 8x(8x^2 - 2x + 1) \]
Since the answer choices provided do not exactly include this factorization, we can also write it in expanded form. However, the most straightforward simplification leads us to options provided.
None of the given answers seem to reflect this directly, but if your response options included \( 8x^2 - 2x + 1 \), that would be part of the expression.
If we look at the options again:
- \( 8x^3 - x^2 + 1 \)
- \( 64x^3 - 16x^2 \)
- \( 0 \)
- \( 8x^2 - 2x + 1 \)
The choice \( 8x^2 - 2x + 1 \) is closest to what we evaluated, but should probably have included the factor of \( 8x \).
So the correct response based solely on simplification looks like this:
Final answer: 8x(8x^2 - 2x + 1), and from given choices, the logically matching one would be the last option \( 8x^2 - 2x + 1 \). If this were to have \( 8x \) in front or be described as equivalent based on context, it could be used here.