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  1. Chapter 2 Class 10 Polynomials
  2. Serial order wise
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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 βˆ’ 5 (1/2) + 2 = 1/4 + 1/4 βˆ’ 5/2 + 2 = (1 + 1 βˆ’ 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 βˆ’ 5 + 2 = 5 βˆ’ 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 = βˆ’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 Ξ±, 𝛽 and Ξ³ We have Ξ± + 𝛽 + Ξ³ = (βˆ’π‘)/π‘Ž Ξ±"𝛽" + 𝛽γ + Ξ³Ξ± = 𝑐/π‘Ž Ξ±"𝛽" Ξ³= (βˆ’π‘‘)/π‘Ž For p(x) = 2x3 + x2 βˆ’ 5x + 2 a = 3, b = βˆ’5, c = βˆ’11 and d = βˆ’3 And zeroes are 𝛼 = 1/2, 𝛽 = 1 and 𝛾 = βˆ’2 Now 𝜢+ 𝜷 + 𝜸 = 1/2 + 1 βˆ’ 2 = (1 + 2 βˆ’ 4)/2 = (βˆ’1)/2 = (βˆ’π‘)/π‘Ž 𝜢𝜷+ 𝜷𝜸 + 𝜸𝜢 = (1/2) (1) + (1) (βˆ’2) + (βˆ’2)(1/2) = 1/2 βˆ’ 2 βˆ’ 1 = (1 βˆ’ 4 βˆ’ 2)/2 = (βˆ’5)/2 = 𝑐/π‘Ž 𝜢𝜷𝜸 = 1/2 Γ— 1 Γ— βˆ’2 = (βˆ’2)/2 = (βˆ’π‘‘)/π‘Ž 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 βˆ’ 4 (2)2 + 5(2) βˆ’ 2 = 8 βˆ’ 4(4)+ 10 – 2 = 18 βˆ’ 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 βˆ’ 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 βˆ’ 5 = 0 Since p(1) = 0 Hence, 1 is a zero of p(x) Verifying relationship between zeroes and coefficients. For p(x) = x3 βˆ’ 4x2 + 5x βˆ’ 2 a = 1, b = βˆ’4, c = 5 and d = βˆ’2 And Zeroes are 𝛼 = 2, 𝛽 = 1 and 𝛾 = 1 𝜢+ 𝜷 + 𝜸 = 2 + 1 + 1 = 4 = (βˆ’(βˆ’4))/1 = (βˆ’π‘)/π‘Ž Hence, the relationship is verified

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