Question

divide and list the remainder only. include the missing terms for the 4th, 3rd and second missing powers. (3m⁵ + m - 1) ÷ (m + 1) hint: rewrite this as (3m⁵ + 0m⁴ + 0m³ + 0m² + m - 1) ÷ (m + 1) options: -5, 12, 3, -8

Explanation:

Step1: Use Polynomial Remainder Theorem

The Polynomial Remainder Theorem states that the remainder of a polynomial \( f(m) \) divided by \( m - a \) is \( f(a) \). Here, we are dividing by \( m + 1 = m - (-1) \), so \( a = -1 \).

Step2: Evaluate \( f(-1) \)

Let \( f(m) = 3m^5 + 0m^4 + 0m^3 + 0m^2 + m - 1 \). Substitute \( m = -1 \) into \( f(m) \):
\[

$$\begin{align*} f(-1)&= 3(-1)^5 + 0(-1)^4 + 0(-1)^3 + 0(-1)^2 + (-1) - 1\\ &= 3(-1) + 0 + 0 + 0 - 1 - 1\\ &= -3 - 1 - 1\\ &= -5 \end{align*}$$

\]

Answer:

-5