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Example 5 (Optional) 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. Let p(x) = 3x3 − 5x2 − 11x − 3 Verifying zeroes At x = 𝟑 p(𝟑) = 3 (3)^3 − 5 (3)^2 − 11 (3) − 3 = 3 (27) − 5 (9) − 33 − 3 = 81 − 45 − 33 − 3 = 81 − 81 = 0 Since p(3) = 0 ∴ 𝟑 is a zero of p(x) At x = −𝟏 p(−1) = 3 (−1)3 − 5 (−1)2 – 11 (−1) − 3 = 3 (−1) − 5 (1) + 11 − 3 = − 3 − 5 + 11 − 3 = − 11 + 11 = 0 Since p(−1) = 0 ∴ −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 ∴ ((−𝟏)/𝟑) is a zero of p(x). Verifying relationship between zeroes and coefficients 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 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 Now 𝜶+ 𝜷 + 𝜸 = 3 + (−1) +((−1)/3) = 3 − 1 − 1/3 = (9 − 3 − 1)/3 = 5/3 = (−𝒃)/𝒂 𝜶𝜷+ 𝜷𝜸 + 𝜸𝜶 = (3) (−1) + (−1) ((−1)/3) + ((−1)/3)(3) = −3 + 1/3 − 1 = (− 9 + 1 − 3)/3 = (−11)/3 = 𝒄/𝒂 𝜶𝜷𝜸 = 3 × (−1) ×((−1)/3) = 3/3 = 1 = (−𝒅)/𝒂 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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.