📖 Chapter 2 — Polynomials (CBSE Class 10 Maths): Detailed notes — types & degree, zeroes, geometrical meaning, and the relationship between zeroes and coefficients.
Chapter 2 — Polynomials
🔢 1. Polynomial & its Degree
A polynomial p(x) = aₙxⁿ + … + a₁x + a₀ (aₙ ≠ 0). The highest power of x is its degree.
| Degree | Name | Form |
|---|
| 1 | Linear | ax + b |
| 2 | Quadratic | ax² + bx + c |
| 3 | Cubic | ax³ + bx² + cx + d |
🎯 2. Zero of a Polynomial
A real number k is a zero of p(x) if p(k) = 0. A polynomial of degree n has at most n zeroes.
📈 3. Geometrical Meaning
The zeroes of p(x) are the x-coordinates of the points where the graph y = p(x) cuts (or touches) the x-axis. So the number of zeroes = number of points where the graph meets the x-axis. (A quadratic graph is a parabola.)
🔗 4. Relationship between Zeroes and Coefficients
For a quadratic ax² + bx + c with zeroes α and β:
α + β = −b/a (sum) · αβ = c/a (product)
✍️ 5. Forming a Quadratic from its Zeroes
If sum = S and product = P of the zeroes, the quadratic is:
x² − Sx + P (or k·(x² − Sx + P) for any k ≠ 0)
🧠 Remember: degree 1/2/3 = linear/quadratic/cubic. Zero → p(k)=0 → graph meets x-axis. Sum of zeroes = −b/a, Product = c/a. Quadratic from zeroes = x² − (sum)x + (product).
⭐ Exam Tips: ① Number of zeroes from a graph (count x-axis cuts). ② Find zeroes by factorising (split the middle term). ③ Verify α+β=−b/a and αβ=c/a. ④ Form a quadratic from given sum & product. ⑤ A double root touches the x-axis at one point.