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Last updated at May 29, 2018 by Teachoo

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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 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 = ( )/ 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 a cubic Polynomial p(x) = ax3 + bx2 + cx + d With zeroes , and We have + + = ( )/ " " + + = / " " = ( )/ For p(x) = 3x3 5x2 11x 3, a = 3, b = 5, c = 11 and d = 3 And zeroes are = 3, = 1 and = ( 1)/3 Now + + = 3 + ( 1) + (( 1)/3) = (9 3 1)/3 = 5/3 = ( ( 5))/3 = ( )/ + + = (3) ( 1) + ( 1) (( 1)/3) + (( 1)/3) (3) = 3 + 1/3 3/3 = ( 9 + 1 3)/3 = ( 11)/3 = / = (3) ( 1) (( 1)/3) = 3/3 = ( ( 3))/3 = ( )/ Hence, the relationship is verified

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.