Polynomials
Polynomials and their zeroes
Find the value of a polynomial and spot its zeroes.
Is it a polynomial? Look at the powers of the variable. Every power of x must be a whole number: 0, 1, 2, 3 … .
- 1/(x – 1) has x downstairs, so x has a negative power: not a polynomial.
- √x + 2 has x to the power ½: not a polynomial.
- √5·x is fine: the root is on the number, not on x.
The degree is the highest power of x. Simplify first: in 5x⁴ – 2x + 3x – 5x⁴ the x⁴ terms cancel, so the degree is 1, not 4.
| Degree | Name | General form | Example |
|---|---|---|---|
| 1 | linear | ax + b | 2x – 3 |
| 2 | quadratic ('quadrate' means square) | ax² + bx + c | x² – 3x – 4 |
| 3 | cubic | ax³ + bx² + cx + d | 2 – x³ |
In each general form a ≠ 0; otherwise the top power would disappear.
The value p(k): put k in place of x and work it out. For p(x) = x² – 3x – 4:
- p(2) = 4 – 6 – 4 = –6
- p(0) = –4
- p(–1) = 1 + 3 – 4 = 0
- p(4) = 16 – 12 – 4 = 0
A zero of p(x) is a number k that makes p(k) = 0. So –1 and 4 are the zeroes of x² – 3x – 4. Watch out: x² – 4 has two zeroes, 2 and –2. Don't forget the negative one.
For a linear polynomial ax + b, solve ak + b = 0: the zero is k = –b/a. The zero of 2x + 3 is –3/2.
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