1. What is a Polynomial?
A polynomial in variable x is an expression of the form anxn + an-1xn-1 + ... + a1x + a0, where coefficients ai are real numbers and n is a non-negative integer.
Degree: The highest power of x with a non-zero coefficient. If an ≠ 0, degree = n. Constant non-zero polynomial has degree 0; zero polynomial has undefined (or −∞) degree.
2. Types of Polynomials (by degree)
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | 5 |
| 1 | Linear | 2x + 3 |
| 2 | Quadratic | x² − 5x + 6 |
| 3 | Cubic | x³ − x² + 2 |
3. Remainder Theorem
If a polynomial f(x) is divided by (x − c), the remainder is f(c).
Example. Find remainder when f(x)=x³−2x²+3x−5 is divided by (x−2).
Compute f(2)=8−8+6−5=1 ⇒ remainder = 1.
4. Factor Theorem
(x − c) is a factor of f(x) iff f(c) = 0. Use this to test roots and factor polynomials.
Example. Show (x−3) is a factor of f(x)=x³−6x²+11x−6.
f(3)=27−54+33−6=0 ⇒ (x−3) is a factor. Divide to factor fully: f(x)=(x−3)(x−2)(x−1).
5. Factorisation Methods
- By factor theorem: Find rational roots c using Rational Root Theorem and test f(c)=0.
- Grouping: Group terms and factor common factors.
- Quadratic formula: For degree 2, use ax²+bx+c = 0 ⇒ x = [−b ± √(b²−4ac)]/(2a).
- Special products: a²−b²=(a−b)(a+b), a³±b³=(a±b)(a²∓ab+b²).
6. Graphs & Behaviour
Polynomial graphs are smooth and continuous. End behaviour depends on degree and leading coefficient:
- Even degree, positive leading coef → both ends up.
- Even degree, negative leading coef → both ends down.
- Odd degree, positive leading coef → left down, right up.
- Odd degree, negative leading coef → left up, right down.
Roots correspond to x-intercepts; multiplicity affects crossing/touching.
7. Solved Examples
Example 1. Factorize x³−4x²−7x+10.
Try integer roots ±1,±2,±5,±10. f(1)=0 ⇒ (x−1) factor. Divide to get (x−1)(x²−3x−10) = (x−1)(x−5)(x+2).
Example 2. If f(x)=2x³+3x²−11x−6 and (x+2) is a factor, factorize fully.
Use synthetic division: dividing by (x+2) gives quadratic 2x²−x−3 → (2x+3)(x−1) ⇒ f(x)=(x+2)(2x+3)(x−1).
8. Practice Questions
- Find remainder when x⁴−3x³+2x−5 is divided by (x−1). (Answer toggle)
- Factorize x³+6x²+11x+6.
- Show that (x−2) is a factor of x³−5x²+8x−4 and factorize fully.
- Find all real roots of 2x³−x²−8x+4.
9. Exam Tips & Tricks
- Test small integer divisors of constant term (±1, ± factors) to find roots quickly.
- Use synthetic division for speed.
- Remember special products and grouping patterns.