Algebra (modulus and polynomials)
The modulus
- The modulus $|x|$ is the size of a number, sign removed ($|x| \geq 0$). Its graph is a "V".
- Useful rules:
- $|a| = |b| \Leftrightarrow a^2 = b^2$,
- $|x - a| < b \Leftrightarrow a - b < x < a + b$.
Practice
Solve |x − 2| < 3. The solution is −1 < x < ? What is the upper bound?
|x − 2| < 3 means 2 − 3 < x < 2 + 3, i.e. −1 < x < 5.
Factor & remainder theorems
- A polynomial is a sum of powers of $x$; its degree is the highest power.
- Remainder theorem: the remainder when $p(x) \div (x - a)$ is $p(a)$.
- Factor theorem: $(x - a)$ is a factor exactly when $p(a) = 0$.
Practice
By the remainder theorem, what is the remainder when p(x) = x² + 3x + 2 is divided by (x − 1)?
Remainder = p(1) = 1 + 3 + 2 = 6.
Practice
(x − a) is a factor of p(x) exactly when:
The factor theorem: (x − a) is a factor if and only if p(a) = 0.
You've got it
Key idea
- $|x|$ removes the sign; $|x - a| < b \Leftrightarrow a - b < x < a + b$
- remainder when $p(x) \div (x-a)$ is $p(a)$
- factor theorem: $(x-a)$ is a factor $\Leftrightarrow p(a) = 0$