Exercises: Apply the Remainder Theorem

The polynomial division algorithm states that for polynomials $p(x)$ and $d(x)$, there exist unique polynomials $q(x)$ and $r(x)$ such that which equation holds?

According to the Factor Theorem, which statement is true about a polynomial $p(x)$ and a number $a$?

When $p(x)$ is divided by $(x - a)$, the Remainder Theorem says the remainder equals which of the following?

Use the Remainder Theorem to find the remainder when $p(x) = x^3 - 2x + 4$ is divided by $(x - 3)$.

Use the Remainder Theorem to find the remainder when $p(x) = x^4 + x^2 - 10$ is divided by $(x + 2)$.

Note: $(x + 2) = (x - (-2))$, so $a = -2$.

Use synthetic division to divide $p(x) = 2x^3 - 3x^2 + x - 5$ by $(x - 2)$.

What is the remainder?

After performing synthetic division of $p(x)$ by $(x - a)$, which number in the bottom row of the synthetic division table is the remainder?

$p(x) = x^3 + 2x^2 - 5x - 6$. Is $(x - 2)$ a factor of $p(x)$?

$p(x) = x^3 - 2x^2 - 5x + 6$. Test whether $(x - 1)$ is a factor by computing $p(1)$.

Enter the value of $p(1)$.