Mathematics | Class 10 Mathematics Notes

2.3 Relationship between Zeroes and Coefficients of a Polynomial

This section explores the algebraic relationships between the zeros of polynomials and their coefficients, starting with quadratic polynomials. For a quadratic polynomial p(x) = ax² + bx + c (a ≠ 0), the zeros α and β satisfy the relations: sum of zeros α + β = -b/a and product of zeros αβ = c/a. This is demonstrated by factorizing quadratic polynomials using the method of splitting the middle term. For example, 2x² - 8x + 6 factorizes as 2(x - 1)(x - 3), giving zeros 1 and 3. Their sum is 4 = -(-8)/2 and product is 3 = 6/2, matching the relations. Another example, 3x² + 5x - 2 factorizes as (3x - 1)(x + 2), zeros 1/3 and -2, sum -5/3 = -b/a, product -2/3 = c/a. The general proof shows that if p(x) = k(x - α)(x - β), then expanding and comparing coefficients yields the relations. Examples verify these relations for polynomials like x² + 7x + 10 and x² - 3. The section also addresses the inverse problem: constructing a quadratic polynomial given sum and product of zeros. For cubic polynomials p(x) = ax³ + bx² + cx + d with zeros α, β, γ, the relations extend to: sum of zeros α + β + γ = -b/a, sum of products of zeros taken two at a time αβ + βγ + γα = c/a, and product of zeros αβγ = -d/a. An example with p(x) = 2x³ - 5x² - 14x + 8 confirms these relations for zeros 4, -2, and 1/2. Another example verifies the relations for zeros 3, -1, and -1/3 of p(x) = 3x³ - 5x² - 11x - 3. These relationships provide a powerful tool to connect polynomial coefficients with their zeros without explicit factorization.

🔗 Connection: Prepares for Exercise 2.2 which involves finding zeros and verifying relationships between zeros and coefficients.