Precalculus / Unit 2: Polynomial and Rational Functions
2.3 Polynomial Division and the Remainder Theorem
Notes
You can divide polynomials by long division, or by synthetic division when the divisor is $x - c$.
- Remainder Theorem: the remainder when $f(x)$ is divided by $x - c$ is $f(c)$.
- Factor Theorem: $x - c$ is a factor of $f(x)$ exactly when $f(c) = 0$.
- Example: divide $x^3 - 2x^2 - 5x + 6$ by $x - 3$. Synthetic division with $c = 3$ on $1, -2, -5, 6$ gives $1, 1, -2$ and remainder $0$, so the quotient is $x^2 + x - 2$.
Practice
What is the remainder when $f(x) = x^3 + 2x - 5$ is divided by $x - 2$?
By the Remainder Theorem, $f(2) = 8 + 4 - 5 = 7$.
Divide $x^3 - 2x^2 - 5x + 6$ by $x - 3$. What is the quotient?
Synthetic division with 3 gives coefficients $1, 1, -2$ and remainder 0.
For what value of $k$ is $x + 1$ a factor of $x^3 + kx - 6$?
We need $f(-1) = -1 - k - 6 = 0$, so $k = -7$.
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