Question

a student divided $f(x) = 3x^3 + 8x^2 + 5x - 4$ by $x + 2$ and found the remainder was $r = -6$. based on the remainder theorem, what can be concluded about $f(x)$? the function $f(x) = 3x^3 + 8x^2 + 5x - 4$ has a zero at $-2$. the function $f(x) = 3x^3 + 8x^2 + 5x - 4$ has a zero at $-6$. the y-intercept of the graph of $f(x)$ is $-6$. the point $(-2, -6)$ lies on the graph of $f(x)$.

Explanation:

Step1: Recall Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). Here, we divide by \( x + 2 = x - (-2) \), so \( a = -2 \), and the remainder \( r = f(-2) = -6 \).

Step2: Analyze Each Option

• Option 1: A zero at \( x = -2 \) would mean \( f(-2) = 0 \), but \( f(-2) = -6

eq 0 \), so this is false.

• Option 2: A zero at \( x = -6 \) would mean \( f(-6) = 0 \), but we know \( f(-2) = -6 \), not related to \( x = -6 \), so false.
• Option 3: The y - intercept is \( f(0) \). Calculate \( f(0)=3(0)^3 + 8(0)^2+5(0)-4=-4

eq -6 \), so false.

• Option 4: Since \( f(-2)=-6 \), by the definition of a function, when \( x = -2 \), \( y = f(-2)=-6 \), so the point \( (-2, -6) \) lies on the graph of \( f(x) \), which is true.

Answer:

The point \((-2, -6)\) lies on the graph of \( f(x) \).