Polynomials – Class 10 Maths Chapter 2 NCERT Notes & Zeroes
Class 10 Polynomials notes: zeroes of polynomial, relationship between zeroes and coefficients, division algorithm for CBSE & ICSE.
Chapter 2: Polynomials
2.1 Geometrical Meaning of Zeroes
The zeroes of a polynomial p(x) are the values of x for which p(x) = 0. Geometrically, they are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
- Linear polynomial (degree 1): ax + b → has exactly 1 zero at x = −b/a
- Quadratic polynomial (degree 2): ax² + bx + c → has at most 2 zeroes
- Cubic polynomial (degree 3): ax³ + bx² + cx + d → has at most 3 zeroes
In general, a polynomial of degree n has at most n zeroes.
2.2 Relationship Between Zeroes and Coefficients
For a Quadratic Polynomial ax² + bx + c (with zeroes α and β)
Product of zeroes: α × β = c/a
The polynomial can be written as: a[x² − (α + β)x + αβ]
Example: Find the zeroes of x² − 5x + 6 and verify the relationship.
x² − 5x + 6 = (x − 2)(x − 3) = 0 → zeroes are α = 2, β = 3
- Sum: α + β = 2 + 3 = 5 = −(−5)/1 = −b/a ✓
- Product: α × β = 2 × 3 = 6 = 6/1 = c/a ✓
For a Cubic Polynomial ax³ + bx² + cx + d (with zeroes α, β, γ)
αβ + βγ + γα = c/a
αβγ = −d/a
2.3 Division Algorithm for Polynomials
If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that:
where degree of r(x) < degree of g(x), or r(x) = 0.
This is used to find the remaining zeroes of a polynomial when some zeroes are already known.
Indexed Topics & Examination Keywords
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