Ex 2.2, 1 (iii) - Chapter 2 Class 10 Polynomials

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Ex 2.2, 1 Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (iii) 6x2 – 3 – 7x Let p(x) = 6x2 – 7x – 3 Zero of the polynomial is the value of x where p(x) = 0 Putting p(x) = 0 6x2 – 7x – 3 = 0 We find roots using splitting the middle term method 6x2 – 9x + 2x – 3 = 0 3x(2x – 3) + 1(2x – 3) = 0 (3x + 1)(2x – 3) = 0 Splitting the middle term method We need to find two numbers where Sum = -7 Product = -3 × 6 = -18 Thus, 3x + 1 = 0 ∴ x = – 𝟏/𝟑 2x – 3 = 0 ∴ x = 𝟑/𝟐 Therefore, α = – 𝟏/𝟑 & β = 𝟑/𝟐 are the zeroes of the polynomial Now, verifying zeroes Verifying relationship b/w zeroes and coefficients p(x) = 6x2 – 7x – 3 Comparing with as2 + bs + c a = 6 , We have to verify Sum of zeroes = − (𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥)/(𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑥2) i.e. α + β = - 𝒃/𝒂 α + β = (−1)/3 + 3/2 = (−1 × 2 + 3 ×3 )/(3 × 2) = (−2 + 9 )/6 = 𝟕/𝟔 − 𝑏/𝑎 = −((−7))/6 = 𝟕/𝟔 α β = (−1)/3 × 3/2 = (−𝟏)/𝟐 𝑐/𝑎 = (−3)/6 = (−𝟏)/𝟐 Since, L.H.S = R.H.S Hence relationship between zeroes & coefficient is verified