Q1. Find the remainder when x³ − 3x² + 4 is divided by (x − 2).
By the remainder theorem, remainder = p(2)
p(2) = 2³ − 3(2²) + 4 = 8 − 12 + 4 = 0
So the remainder is 0 (and x − 2 is a factor).
Polynomials — Class 9
Polynomials factor beautifully once you know the identities and the factor theorem. Master both for the exam.
Polynomial & degreeAn expression like 2x³ − x + 5. The highest power of x is the degree.
Remainder theoremIf p(x) is divided by (x − a), the remainder is p(a).
Factor theorem(x − a) is a factor of p(x) if and only if p(a) = 0.
Algebraic identities(a+b)² = a²+2ab+b², (a−b)² = a²−2ab+b², a²−b² = (a+b)(a−b), and more.
Solved examples
Q2. Factorise x² + 5x + 6.
Find two numbers with product 6 and sum 5: 2 and 3
x² + 5x + 6 = (x + 2)(x + 3)
Formula sheet
| Quantity | Formula |
|---|---|
| Square of a sum | (a+b)² = a² + 2ab + b² |
| Difference of squares | a² − b² = (a+b)(a−b) |
| Sum of cubes | a³ + b³ = (a+b)(a²−ab+b²) |
| (a+b+c)² | a²+b²+c²+2(ab+bc+ca) |
Common mistakes & exam wins
- Remember: remainder when dividing by (x − a) is just p(a).
- (x − a) is a factor exactly when p(a) = 0 (factor theorem).
- Learn the identities cold — they turn hard factorisations into one line.
Frequently asked questions
What is the remainder theorem?
When a polynomial p(x) is divided by (x − a), the remainder equals p(a).
What is the factor theorem?
(x − a) is a factor of p(x) if and only if p(a) = 0.
What is the degree of a polynomial?
The highest power of the variable in the polynomial.