# Ex 2.4, 1 (Optional)

Last updated at Aug. 1, 2017 by Teachoo

Last updated at Aug. 1, 2017 by Teachoo

Transcript

Ex 2.4, 1 Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case: (i) 2x3 + x2 – 5x + 2; 1/2 , 1, – 2 p(x) = 2x3 + x2 – 5x + 2 Verifying zeroes At x = 𝟏/𝟐 p(1/2) = 2 (1/2)^3 + (1/2)^2 − 5 (1/2) + 2 = 1/4 + 1/4 − 5/2 + 2 = (1 + 1 − 10 + 8)/4 = 0/4 = 0 Since p(1/2) = 0 Hence, 1/2 is a zero of p(x) At x = 𝟏 p(1) = 2(1)3 + (1)2 – 5(1) + 2 = 2 + 1 − 5 + 2 = 5 − 5 = 0 Since p(1) = 0 Hence, 1 is a zero of p(x) At x = –2 p(-2) = 2(-2)3 + (-2)2 – 5(-2) + 2 = 16 + 4 + 10 + 2 = −16 + 16 = 0 Since p(-2) = 0 Hence, –2 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) = 2x3 + x2 − 5x + 2 a = 3, b = −5, c = −11 and d = −3 And zeroes are 𝛼 = 1/2, 𝛽 = 1 and 𝛾 = −2 Now 𝜶+ 𝜷 + 𝜸 = 1/2 + 1 − 2 = (1 + 2 − 4)/2 = (−1)/2 = (−𝑏)/𝑎 𝜶𝜷+ 𝜷𝜸 + 𝜸𝜶 = (1/2) (1) + (1) (−2) + (−2)(1/2) = 1/2 − 2 − 1 = (1 − 4 − 2)/2 = (−5)/2 = 𝑐/𝑎 𝜶𝜷𝜸 = 1/2 × 1 × −2 = (−2)/2 = (−𝑑)/𝑎 Hence, the relationship is verified Ex 2.4, 1 Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case: (ii) x3 - 4x2 + 5x – 2; 2, 1, 1 p(x) = x3 - 4x2 + 5x – 2 Verifying zeroes At x = 2 p(2) = (2)3 − 4 (2)2 + 5(2) − 2 = 8 − 4(4)+ 10 – 2 = 18 − 18 = 0 Since p(2) = 0 Hence, 2 is a zero of p(x) At x = 𝟏 p(1) = (1)3 – 4(1)2 + 5(1) – 2 = 1 – 4 + 5 – 2 = 5 − 5 = 0 Since p(1) = 0 Hence, 1 is a zero of p(x) At x = 𝟏 p(1) = (1)3 – 4(1)2 + 5(1) – 2 = 1 – 4 + 5 – 2 = 5 − 5 = 0 Since p(1) = 0 Hence, 1 is a zero of p(x) Verifying relationship between zeroes and coefficients. For p(x) = x3 − 4x2 + 5x − 2 a = 1, b = −4, c = 5 and d = −2 And Zeroes are 𝛼 = 2, 𝛽 = 1 and 𝛾 = 1 𝜶+ 𝜷 + 𝜸 = 2 + 1 + 1 = 4 = (−(−4))/1 = (−𝑏)/𝑎 Hence, the relationship is verified

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CA Maninder Singh

CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .