Definitions [4]
A polynomial is an algebraic expression made up of terms in which the variables have non‑negative whole-number exponents.
When an algebraic expression is made of only one variable, it is called a polynomial in one variable.
Examples of Polynomials in One Variable:
| Polynomial | Variable | Why it’s a polynomial |
|---|---|---|
| 3 + 5x − 7x2 | x | All exponents (0, 1, 2) are whole numbers |
| 9y3 − 5y2 + 8 | y | All exponents (3, 2, 0) are whole numbers |
| z4 + z - 1 | z | All exponents (4, 1, 0) are whole numbers |
The degree of a polynomial is simply the highest exponent (power) in the expression.
Example 1: 4x² - 3x⁵ + 8x⁶
- Term 1: 4x² → exponent = 2
- Term 2: -3x⁵ → exponent = 5
- Term 3: 8x⁶ → exponent = 6
- Degree = 6 (highest exponent)
Example 2: 25 - x⁴
- Term 1: 25 → exponent = 0 (since 25 = 25x⁰)
- Term 2: -x⁴ → exponent = 4
- Degree = 4
A real number k is a zero of p(x) if p(k) = 0.
Formulae [1]
For
p(x) = ax + b
Zero:
Theorems and Laws [1]
Theorem :If p(x) is a polynomial of degree `n >= 1` and a is any real number, then
(i) x – a is a factor of p(x), if p(a) = 0, and
(ii) p(a) = 0, if x – a is a factor of p(x).
Proof: By the Remainder Theorem, p(x)=(x – a) q(x) + p(a).
(i) If p(a) = 0, then p(x) = (x – a) q(x), which shows that x – a is a factor of p(x).
(ii) Since x – a is a factor of p(x), p(x) = (x – a) g(x) for same polynomial g(x).
In this case, p(a) = (a – a) g(a) = 0.
Key Points
Statement:
If a polynomial f(x) is divided by (x − a), then the remainder is f(a).
Result:
Remainder = f(a)
