📐 Polynomials - Class 10

1. Geometrical Meaning of the Zeroes of a Polynomial

The zeroes of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.

• A linear polynomial has exactly one zero.
• A quadratic polynomial has at most two zeroes.
• A cubic polynomial has at most three zeroes.

2. Relationship between Zeroes and Coefficients of a Polynomial

For a quadratic polynomial ax² + bx + c, if α and β are the zeroes, then:

• Sum of zeroes (α + β) = -b/a
• Product of zeroes (αβ) = c/a

For a cubic polynomial ax³ + bx² + cx + d, if α, β and γ are the zeroes, then:

• Sum of zeroes (α + β + γ) = -b/a
• Sum of the products of the zeroes taken two at a time (αβ + βγ + γα) = c/a
• Product of zeroes (αβγ) = -d/a

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 p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).