Question

what is the remainder when ( p(x) ) is divided by ( x - 1 )? ( p(x)=x^{3}-3x^{2}-10x + 24 )
a. 24
b. 12
c. 30
d. 0

Explanation:

Step1: Recall the Remainder Theorem

The Remainder Theorem states that if a polynomial \( p(x) \) is divided by \( x - a \), the remainder is \( p(a) \). Here, we are dividing \( p(x)=x^{3}-3x^{2}-10x + 24 \) by \( x - 1 \), so \( a = 1 \).

Step2: Substitute \( x = 1 \) into \( p(x) \)

Calculate \( p(1) \) by substituting \( x = 1 \) into the polynomial:
\[

$$\begin{align*} p(1)&=(1)^{3}-3(1)^{2}-10(1)+24\\ &=1 - 3 - 10 + 24\\ &=(1 + 24)-(3 + 10)\\ &=25 - 13\\ &=12 \end{align*}$$

\]

Answer:

B. 12