Assessment: The Remainder Theorem and the Factor Theorem

According to the remainder theorem, what does P(a) equal when P(x) is divided by (x−a)?

What does the factor theorem tell you about P(x) when P(a) = 0?

Find the remainder when P(x) = x³ + 2x² − x + 5 is divided by (x − 1).

Use synthetic division to divide x² − 5x + 6 by (x − 2). What is the remainder?

Which value of a makes (x − a) a factor of P(x) = x³ − x² − 4x + 4?

Fully factor x³ − 3x² − x + 3. (Hint: try a = 1 first.)

An engineer has a cost formula that is a cubic polynomial divided by a linear expression. Why would they use polynomial long division?

A proposed break-even quantity for a cubic profit model P(x) = x³ − 6x² + 11x − 6 is x = 2. Confirm by finding P(2). (0 would confirm it is a break-even point.)

P(x) = x³ + kx − 4 leaves a remainder of 5 when divided by (x − 1). Find k.

Is x = 2 a root of P(x) = x³ − 2x² − 5x + 6?