To solve the problem based on the given function [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex] and the information that when divided by [tex]\( x+2 \)[/tex] the remainder is [tex]\(-6\)[/tex], we can use the Remainder Theorem to understand the function further.
The Remainder Theorem states that if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - a \)[/tex], then the remainder of this division is [tex]\( f(a) \)[/tex]. In this problem, the divisor is [tex]\( x + 2 \)[/tex], which can be rewritten as [tex]\( x - (-2) \)[/tex]. Thus, according to the Remainder Theorem, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 2 \)[/tex] is [tex]\( f(-2) \)[/tex].
Given that the remainder is -6, we have:
[tex]\[ f(-2) = -6 \][/tex]
Based on this, let's evaluate each statement:
1. The function [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex] has a zero at -2.
For [tex]\( x = -2 \)[/tex] to be a zero, [tex]\( f(-2) \)[/tex] must equal 0. However, we know from the problem that [tex]\( f(-2) = -6 \)[/tex], not 0. Therefore, -2 is not a zero of the function.
2. The function [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex] has a zero at -6.
This statement is incorrect because zeros are the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex]. The statement is referencing an incorrect concept.
3. The point [tex]\((-2, -6)\)[/tex] lies on the graph of [tex]\( f(x) \)[/tex].
We know from the Remainder Theorem that [tex]\( f(-2) = -6 \)[/tex]. This means that when [tex]\( x = -2 \)[/tex], the output of the function [tex]\( f(x) \)[/tex] is -6. Therefore, the point [tex]\((-2, -6)\)[/tex] does indeed lie on the graph of [tex]\( f(x) \)[/tex].
4. The [tex]\( y \)[/tex]-intercept of the graph of [tex]\( f(x) \)[/tex] is -6.
The [tex]\( y \)[/tex]-intercept is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex]. To find the [tex]\( y \)[/tex]-intercept, we would evaluate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 3(0)^3 + 8(0)^2 + 5(0) - 4 = -4 \][/tex]
The [tex]\( y \)[/tex]-intercept is -4, not -6.
Therefore, the correct statement is:
The point [tex]\((-2, -6)\)[/tex] lies on the graph of [tex]\( f(x) \)[/tex].
