Difficult Polynomial Questions

Chapter 2 Class 10 Polynomials
Serial order wise

### Transcript

Question 1 - Polynomials Class 10 Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeros and the coefficients in each case: (i) 2x^3 + x^2 - 5x + 2; 1/2, 1, -2 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 ∴ 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 ∴ 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 ∴ –2 is a zero of p(x). 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 ∴ 𝟏/𝟐 is a zero of p(x) Verifying relationship between zeroes and coefficients For p(x) = 2x3 + x2 − 5x + 2 a = 2, b = 1, c = −5 and d = 2 And zeroes are 𝜶 = 1/2, 𝜷 = 1 and 𝜸 = −2 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 Now 𝜶+ 𝜷 + 𝜸 = 1/2 + 1 − 2 = (1 + 2 − 4)/2 = (−1)/2 = (−𝒃)/𝒂

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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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.