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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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Example 5 Verify that 3, –1, (−1)/3 are the zeroes of the cubic polynomial p(x) = 3x3 – 5x2 – 11x – 3, and then verify the relationship between the zeroes and the coefficients. p(x) = 3x3 − 5x2 − 11x − 3 Verifying zeroes At x = 3, p(3) = 3 (3)3 − 5(3)2 − 11(3) − 3 = 81 − 45 − 33 − 3 = 0 Since p(3) = 0 Hence, 3 is a zero of p(x) At x = −1, p(−1) = 3 (-1)3 − 5(-1)2 − 11 (-1) − 3 = −3 − 5 + 11 − 3 = 0 Since p(−1) = 0 Hence, −1 is a zero of p(x) At x = (−𝟏)/𝟑 p((−1)/3) = 3((−1)/3)^3− 5((−1)/3)^2 − 11 ((−1)/3) − 3 = (−1)/9 − 5/9 + 11/3 − 3 = (−1 − 5 + 33 − 27)/9 = (−33+33)/9 = 0 Since p((−𝟏)/𝟑) = 0. Hence, (−𝟏)/𝟑 is a zero of P (x) Verifying relationship between zeroes and coefficients. For p(x) = 3x3 − 5x2 − 11x − 3, a = 3, b = −5, c = −11 and d = −3 And zeroes are 𝜶 = 3, 𝛽 = −1 and 𝜸 = (−𝟏)/𝟑 For a cubic Polynomial p(x) = ax3 + bx2 + cx + d With zeroes α, 𝛽 and γ We have α + 𝛽 + γ = (−𝑏)/𝑎 α"𝛽" + 𝛽γ + γα = 𝑐/𝑎 α"𝛽" γ= (−𝑑)/𝑎 𝜶+ 𝜷 + 𝜸 = 3 + (−1) + ((−1)/3) = (9 − 3 − 1)/3 = 5/3 = (−(−𝟓))/𝟑 = (−𝒃)/𝒂 𝜶𝜷+ 𝜷𝜸 + 𝜸𝜶 = (3) (−1) + (−1) ((−1)/3) + ((−1)/3) (3) = –3 + 1/3 – 3/3 = (−9 + 1 −3)/3 = (−𝟏𝟏)/𝟑 = 𝒄/𝒂 𝜶𝜷𝜸 = (3) × (− 1) × ((−1)/3) = 3/3 = (−(−𝟑))/𝟑 = (−𝒅)/𝒂 Now Hence, the relationship is verified

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.