Question

1. if the polynomial $p(x)=x^3 + x$ is divided by $x + 3$, what is the remainder?

Explanation:

Step1: Recall Remainder Theorem

The Remainder Theorem states that if a polynomial \( p(x) \) is divided by \( x - a \), the remainder is \( p(a) \). For \( x + 3 \), we can rewrite it as \( x - (-3) \), so \( a = -3 \).

Step2: Evaluate \( p(-3) \)

Given \( p(x) = x^3 + x \), substitute \( x = -3 \) into the polynomial:
\( p(-3) = (-3)^3 + (-3) \)
Calculate \( (-3)^3 = -27 \), then \( -27 + (-3) = -30 \).

Answer:

The remainder is \(-30\).