Increase the roots of the equation x4 – 2x3 + 14x -15 = 0 by 5.
Answers
Given : Increase the roots of the equation x³– 2x² + 14x -15 = 0 by 5.
To Find : New Equation
Solution:
x³– 2x² + 14x -15 = 0
Roots are
a , b , c
a + b + c = -(-2)/1 = 2
ab + bc + ac = 14/1 = 14
abc = -(-15)/1 = 15
roots increased by 5
a + 5 , b + 5 , c + 5
sum of roots = a + b + c + 15 = 2 + 15 = 17
(a + 5)(b + 5) + (a + 5)(c + 5) + (c + 5)(b + 5)
= ab + 5(a + b) + 25 +ac + 5(a + c) + 25 + bc + 5(b + c) + 25
= ab + bc + ac + 10(a + b + c) + 75
= 14 + 10(2) + 75
= 109
(a + 5)(b + 5)(c + 5) = (ab + 5a + 5b + 25)(c + 5)
= abc + 5ac + 5bc + 25c + 5ab + 25a + 25b + 125
= abc + 5(ab + bc + ac) + 25(a + b + c) + 125
= 15 + 5(14) + 25(2) + 125
= 15 + 70 + 50 + 125
= 260
x³ - 17x² + 109x - 260 = 0
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