Types of Polynomials by Degree
| Degree | Name | General form | Max zeroes |
|---|---|---|---|
| 0 | Constant | k | 0 |
| 1 | Linear | ax + b | 1 |
| 2 | Quadratic | ax² + bx + c | 2 |
| 3 | Cubic | ax³ + bx² + cx + d | 3 |
Zero of a polynomial p(x): A value k such that p(k) = 0.
Geometrical Meaning of Zeroes
The zeroes of a polynomial y = p(x) are the x-coordinates where the graph crosses or touches the x-axis.
Quadratic polynomial (parabola):
- Intersects x-axis at 2 points → 2 real distinct zeroes
- Touches x-axis at 1 point → 2 equal zeroes (repeated root)
- Does not intersect x-axis → no real zeroes
Relationship Between Zeroes and Coefficients
Quadratic Polynomial: ax² + bx + c
If α and β are the zeroes:
| Relation | Formula |
|---|---|
| Sum of zeroes | α + β = −b/a |
| Product of zeroes | αβ = c/a |
Forming a quadratic polynomial from its zeroes: x² − (sum of zeroes)x + (product of zeroes) = 0 x² − (α + β)x + αβ = 0
Worked example: Find a quadratic polynomial whose zeroes are 3 and −5.
- Sum = 3 + (−5) = −2
- Product = 3 × (−5) = −15
- Polynomial: x² − (−2)x + (−15) = x² + 2x − 15
Cubic Polynomial: ax³ + bx² + cx + d
If α, β, γ are the zeroes:
| Relation | Formula |
|---|---|
| α + β + γ | = −b/a |
| αβ + βγ + γα | = c/a |
| αβγ | = −d/a |
Finding Zeroes
Method 1 — Factorisation
p(x) = x² − 5x + 6 = (x − 2)(x − 3) Zeroes: x = 2 and x = 3
Verify: Sum = 2 + 3 = 5 = −(−5)/1 ✓ ; Product = 2 × 3 = 6 = 6/1 ✓
Method 2 — Using One Known Zero
If one zero of p(x) = x² − 3x + k is 2, find k and the other zero.
Since 2 is a zero: p(2) = 0 4 − 6 + k = 0 → k = 2
Sum of zeroes = 3, one zero = 2, so other zero = 1
Division Algorithm for Polynomials
p(x) = g(x) × q(x) + r(x)
Where degree(r) < degree(g), or r(x) = 0.
Worked example: Divide p(x) = x³ − 3x² + 5x − 3 by g(x) = x² − 2.
Long division gives: q(x) = x − 3, r(x) = 7x − 9
Verify: g(x) × q(x) + r(x) = (x² − 2)(x − 3) + (7x − 9) = x³ − 3x² − 2x + 6 + 7x − 9 = x³ − 3x² + 5x − 3 ✓
Using Division Algorithm to Find Zeroes
If you know one zero of a cubic, divide the cubic by (x − zero) to get a quadratic, then factorise the quadratic to find remaining zeroes.
Worked example: Two zeroes of p(x) = x⁴ − 6x³ − 26x² + 138x − 35 are 2 ± √3.
Factors corresponding to these zeroes: [x − (2+√3)][x − (2−√3)] = (x−2)² − 3 = x² − 4x + 1
Divide p(x) by (x² − 4x + 1) → get x² − 2x − 35 = (x − 7)(x + 5)
All four zeroes: 2+√3, 2−√3, 7, −5
Board Exam Tips
- Zeroes and coefficients — 2–3 mark questions appear every year; learn the formulae
- Form a polynomial from given zeroes — use sum and product formulae
- Division algorithm — mostly 3–4 mark numericals; learn long division step by step
- Always verify your answer by checking sum and product of zeroes
- Cubic polynomial zeroes sum: common trap is forgetting the sign — α+β+γ = −b/a (negative b over a)
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