Let’s analyze the given statements and determine who is correct, Jeff or Annie.
First, consider Jeff's statement:
- Jeff claims that [tex]\(-8x^4\)[/tex] is a factor of [tex]\(4x^3\)[/tex].
To determine if one expression is a factor of another, the factor must divide the other expression exactly. In simpler terms, if [tex]\(-8x^4\)[/tex] is a factor, there must be an integer [tex]\(k\)[/tex] such that [tex]\( (4x^3) \cdot k = -8x^4 \)[/tex].
To verify this, we need to see if we can find such [tex]\(k\)[/tex]:
[tex]\[
\frac{4x^3}{-8x^4} = \frac{4}{-8} \cdot \frac{x^3}{x^4} = -\frac{1}{2x}.
\][/tex]
The result is [tex]\(-\frac{1}{2x}\)[/tex], which suggests that [tex]\(4x^3\)[/tex] multiplied by [tex]\(-2x\)[/tex] results in [tex]\((-8)x^4\)[/tex]:
Thus,
[tex]\[ 4x^3 \cdot (-2x) = -8x^4. \][/tex]
However, a factor must be less than or equal to the number it divides, and in polynomial terms, the degree of the factor must be less than or equal to the degree of the polynomial it divides. The degree of [tex]\(-8x^4\)[/tex] (which is 4) is not less than or equal to the degree of [tex]\(4x^3\)[/tex] (which is 3). Thus, mathematically speaking, Jeff is incorrect because [tex]\( -8x^4 \)[/tex] cannot be a factor of [tex]\(4 x^3\)[/tex].
Now, let's consider Annie's statement:
- Annie claims that [tex]\( -8x^4 \)[/tex] is divisible by [tex]\( 4x^3 \)[/tex].
To check this, we divide [tex]\(-8x^4\)[/tex] by [tex]\(4x^3\)[/tex]:
[tex]\[
\frac{-8x^4}{4x^3} = -2x.
\][/tex]
The result is a valid algebraic expression (-2x), meaning that \ -8x^4\ is indeed divisible by [tex]\(4x^3\)[/tex].
Based on this analysis:
- Jeff is incorrect in his assertion.
- Annie is correct in her assertion.
Therefore, the correct answer is:
B Annie
