To understand the given problem, let's use the Remainder Theorem and analyze the function [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex].
### Step-by-Step Solution:
1. Understanding the Remainder Theorem:
- The Remainder Theorem states that for a polynomial [tex]\( f(x) \)[/tex], if you divide [tex]\( f(x) \)[/tex] by [tex]\( x - a \)[/tex], the remainder of this division is [tex]\( f(a) \)[/tex].
- In the given problem, the polynomial [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex] is divided by [tex]\( x + 2 \)[/tex], which can be rewritten as [tex]\( x - (-2) \)[/tex].
- Therefore, according to the Remainder Theorem, [tex]\( f(-2) \)[/tex] should be the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 2 \)[/tex].
2. Given Information:
- The remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 2 \)[/tex] is [tex]\( -6 \)[/tex]. This implies that:
[tex]\[
f(-2) = -6
\][/tex]
3. Interpreting [tex]\( f(-2) = -6 \)[/tex]:
- The point [tex]\( (-2, -6) \)[/tex] lies on the graph of [tex]\( f(x) \)[/tex]. This is because when [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -6 \)[/tex].
4. Checking Potential Statements:
- The point [tex]\((-2,-6)\)[/tex] lies on the graph of [tex]\( f(x) \)[/tex]:
- Yes, this is true as explained above.
- The function [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex] has a zero at -2:
- A zero of the function occurs when [tex]\( f(x) = 0 \)[/tex]. Since [tex]\( f(-2) = -6 \)[/tex], [tex]\( -2 \)[/tex] is not a zero of the function. This statement is incorrect.
- The function [tex]\( f(x) = 3x^3 + 8x^2 + 5x - 4 \)[/tex] has a zero at -6:
- This statement is a misunderstanding of what a zero is. Zeros are the roots where [tex]\( f(x) = 0 \)[/tex]. This statement is also incorrect.
- The [tex]\( y \)[/tex]-intercept of the graph of [tex]\( f(x) \)[/tex] is [tex]\(-6\)[/tex]:
- The [tex]\( y \)[/tex]-intercept is found by evaluating [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3(0)^3 + 8(0)^2 + 5(0) - 4 = -4
\][/tex]
- Hence, the [tex]\( y \)[/tex]-intercept is [tex]\( -4 \)[/tex], not [tex]\( -6 \)[/tex]. This statement is incorrect.
### Conclusion:
From the problems and solution steps, the correct conclusion based on the evaluation is:
"The point [tex]\( (-2, -6) \)[/tex] lies on the graph of [tex]\( f(x) \)[/tex]."
