To solve this problem, we'll use the Remainder Theorem, which states that if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - a \)[/tex], the remainder is [tex]\( f(a) \)[/tex].
Given the polynomial [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex] and the divisor [tex]\( x + 2 \)[/tex], which can be rewritten as [tex]\( x - (-2) \)[/tex], we need to evaluate [tex]\( f(-2) \)[/tex].
Calculating [tex]\( f(-2) \)[/tex]:
Since the student found the remainder to be [tex]\( -6 \)[/tex] when dividing [tex]\( f(x) \)[/tex] by [tex]\( x + 2 \)[/tex], by the Remainder Theorem, we know:
[tex]\[ f(-2) = -6 \][/tex]
Now, let's interpret the given multiple-choice statements:
1. "The point [tex]\((-2, -6)\)[/tex] lies on the graph of [tex]\( f(x) \)[/tex]."
This is true. Since [tex]\( f(-2) = -6 \)[/tex], the point [tex]\((-2, -6)\)[/tex] is indeed on the graph of [tex]\( f(x) \)[/tex].
2. "The function [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex] has a zero at -2."
This is false. A zero of a function [tex]\( f(x) \)[/tex] implies [tex]\( f(a) = 0 \)[/tex]. In this case, [tex]\( f(-2) = -6 \)[/tex], not 0, so [tex]\(-2\)[/tex] is not a zero of the function.
3. "The [tex]\( y \)[/tex]-intercept of the graph of [tex]\( f(x) \)[/tex] is -6."
This is false. The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. We need to evaluate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 3(0)^3 + 8(0)^2 + 5(0) - 4 = -4 \][/tex]
So the [tex]\( y \)[/tex]-intercept is at [tex]\((0, -4)\)[/tex], not -6.
4. "The function [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex] has a zero at 6."
This is false. To determine if 6 is a zero of [tex]\( f(x) \)[/tex], we would need [tex]\( f(6) = 0 \)[/tex]. This specific evaluation is not provided and is not directly relevant to the problem involving division by [tex]\( x + 2 \)[/tex], but from the given information, we know 6 is not the root related to the remainder provided.
Accordingly, the correct interpretation is:
The point [tex]\((-2, -6)\)[/tex] lies on the graph of [tex]\( f(x) \)[/tex].
