Polynomials, Zeroes and Factorisation: Class 9 Study Guide
A polynomial is a sum of terms with whole-number powers of a variable. Zeroes and factors are closely linked.
A polynomial in x has terms like 3x², −5x and 7. The highest power of x is its degree. A polynomial of degree 1 is linear, degree 2 is quadratic and degree 3 is cubic. A zero of p(x) is a number a with p(a) = 0.
For p(x) = x² − 5x + 6 we find p(2) = 4 − 10 + 6 = 0 and p(3) = 9 − 15 + 6 = 0, so 2 and 3 are its zeroes.
Remainder and factor theorems
The remainder theorem says that when p(x) is divided by (x − a), the remainder is p(a). The factor theorem says that (x − a) is a factor of p(x) exactly when p(a) = 0. For p(x) = x³ − 2x² + x − 5, dividing by (x − 1) leaves p(1) = 1 − 2 + 1 − 5 = −5. Since p(2) = 4 − 10 + 6 = 0 for x² − 5x + 6, (x − 2) is a factor.
From (x − a).
Find p(a).
(x − a) is a factor.
p(a) is the remainder.
Identities
| Identity | Example |
|---|---|
| (x + y)² = x² + 2xy + y² | 103² = (100 + 3)² = 10609 |
| (x − y)² = x² − 2xy + y² | 99² = (100 − 1)² = 9801 |
| x² − y² = (x + y)(x − y) | 99 × 101 = 100² − 1 = 9999 |
| (x + a)(x + b) = x² + (a + b)x + ab | x² + 7x + 12 = (x + 3)(x + 4) |
| x³ + y³ = (x + y)(x² − xy + y²) | 8 + 27 = 35 = 5 × 7 |
Factorising
First take out any common factor. Then look for an identity, or split the middle term. For x² + 7x + 12, find two numbers with product 12 and sum 7: 3 and 4. So x² + 7x + 12 = (x + 3)(x + 4). Always check by multiplying back.
Worked examples and common mistakes
Example: Factorise x² − 9 = (x + 3)(x − 3). Example: Use an identity to find 98 × 102 = 100² − 2² = 9996. Common mistakes: writing (x + y)² = x² + y², missing the middle term, and thinking that the degree is the number of terms. Degree is the highest power.
Check your understanding: model answers
1. It factorises as (x − 3)(x − 4), so the zeroes are 3 and 4. 2. p(1) = 1 + 3 + 2 = 6. 3. 4x² + 12x + 9.
✨ Key Takeaways
- Degree is the highest power of the variable.
- A zero of p(x) satisfies p(a) = 0.
- (x − a) is a factor if and only if p(a) = 0.
- Learn the identities and check by multiplying back.
💬 Questions for Discussion & Reflection
- Find the zeroes of x² − 7x + 12.
- Find the remainder when x² + 3x + 2 is divided by (x − 1).
- Expand (2x + 3)².
Print two-step equation worksheets →