Integrated Math 3 / Unit 1: Polynomial Functions
1.5 Remainder and Factor Theorems
Notes
- Remainder Theorem: when $f(x)$ is divided by $x - c$, the remainder is $f(c)$.
- Factor Theorem: $x - c$ is a factor of $f(x)$ exactly when $f(c) = 0$.
- Example: $f(x) = x^3 - 4x^2 + x + 6$ has $f(2) = 8 - 16 + 2 + 6 = 0$, so $x - 2$ is a factor.
- Once you find one zero, divide it out and factor the quotient to find the rest.
Practice
What is the remainder when $f(x) = 2x^3 + x - 5$ is divided by $x - 1$?
By the Remainder Theorem, $f(1) = 2 + 1 - 5 = -2$.
For which value of $k$ is $x + 2$ a factor of $f(x) = x^3 + kx + 10$?
We need $f(-2) = -8 - 2k + 10 = 2 - 2k = 0$, so $k = 1$.
The polynomial $x^3 - 6x^2 + 11x - 6$ has $x - 1$ as a factor. What are all its zeros?
Dividing by $x - 1$ gives $x^2 - 5x + 6 = (x - 2)(x - 3)$.
Ready for more? Take the Unit 1 test.