Chapter 2: Polynomials
1. What is a Polynomial?
A polynomial in one variable x is an algebraic expression of the form p(x) = anxn + an−1xn−1 + ... + a1x + a0, where a0, a1, ..., an are real numbers (coefficients), an ≠ 0, and all powers of x are whole numbers (0, 1, 2, ...).
- Expressions with negative or fractional powers of x are not polynomials. E.g., x + 1/x, √x + 3 are not polynomials.
- Degree of a polynomial = highest power of the variable with non-zero coefficient.
2. Types of Polynomials by Degree
| Name | Degree | General form | Example |
|---|---|---|---|
| Linear | 1 | ax + b, a ≠ 0 | 2x − 3 |
| Quadratic | 2 | ax2 + bx + c, a ≠ 0 | x2 − 5x + 6 |
| Cubic | 3 | ax3 + bx2 + cx + d, a ≠ 0 | x3 − 4x |
A non-zero constant (e.g., 7) is a polynomial of degree 0. The zero polynomial (0) has its degree not defined.
3. Value and Zero of a Polynomial
The value of p(x) at x = k is p(k), obtained by substituting x = k.
A real number k is a zero of p(x) if p(k) = 0.
Micro-example: For p(x) = x2 − 3x − 4, p(4) = 16 − 12 − 4 = 0, so 4 is a zero. Also p(−1) = 1 + 3 − 4 = 0, so −1 is a zero.
4. Geometrical Meaning of Zeroes
The zeroes of p(x) are the x-coordinates of the points where the graph y = p(x) intersects the x-axis.
- A linear polynomial has exactly 1 zero (its graph is a straight line cutting the x-axis at one point).
- A quadratic polynomial has at most 2 zeroes. Its graph is a parabola — opening upward if a > 0, downward if a < 0.
- A polynomial of degree n has at most n zeroes.
5. Relationship Between Zeroes and Coefficients (Quadratic)
If α and β are the zeroes of the quadratic polynomial p(x) = ax2 + bx + c, a ≠ 0, then:
Product of zeroes: αβ = c/a = (constant term)/(coefficient of x2)
Micro-example: For x2 − 5x + 6 = (x − 2)(x − 3): zeroes are 2 and 3. Check: 2 + 3 = 5 = −(−5)/1 ✓; 2 × 3 = 6 = 6/1 ✓.
6. Forming a Quadratic Polynomial from its Zeroes
If the sum of zeroes is S and the product is P, then a quadratic polynomial is:
Micro-example: Zeroes √2 and −√2 ⇒ S = 0, P = −2 ⇒ polynomial: x2 − 2.
7. Finding Zeroes by Factorisation (Splitting the Middle Term)
- Write p(x) = ax2 + bx + c. Find two numbers whose product = ac and sum = b.
- Split bx using these numbers, factor by grouping.
- Set each factor equal to 0 to get the zeroes, then verify the relationships α + β = −b/a and αβ = c/a.
Worked example: Find zeroes of 6x2 − 7x − 3.
ac = −18, b = −7 ⇒ numbers: −9 and 2.
6x2 − 9x + 2x − 3 = 3x(2x − 3) + 1(2x − 3) = (2x − 3)(3x + 1).
Zeroes: x = 3/2 and x = −1/3.
Verification: sum = 3/2 − 1/3 = 7/6 = −(−7)/6 ✓; product = (3/2)(−1/3) = −1/2 = −3/6 ✓.
8. Useful Identities for Symmetric Expressions of Zeroes
Board questions often ask for expressions in α, β without finding the zeroes themselves. Memorise:
(α − β)2 = (α + β)2 − 4αβ
α3 + β3 = (α + β)3 − 3αβ(α + β)
1/α + 1/β = (α + β)/αβ ; α/β + β/α = (α2 + β2)/αβ
9. Common Board Question Patterns
- Find zeroes of a given quadratic and verify the relationship with coefficients (2–3 marks).
- Form a quadratic polynomial given sum and product of zeroes, or given the zeroes (1–2 marks).
- If α, β are zeroes of a given polynomial, evaluate expressions like α2 + β2, 1/α + 1/β (2–3 marks).
- Find the value of an unknown k when one zero is given, or when zeroes satisfy a condition (e.g., one zero is the negative/reciprocal of the other) (2–3 marks).
- Graph-based MCQ/1-mark: number of zeroes from a graph; shape of parabola.