If t^2-1 is a factor of at^3+t^2-2t+b.Find the value of a and b?
Answers
as t²-1 is a factor so for t=1 and t=-1 ;so h= at³+t²-2 t +b =0
for t=1 ;h=a+1-2+b=0→h=a-1+b=0
for t=-1;h=-a+1+2+b=0→h=-a+3+b=0 solving these two equations we get a=2 and b=-1
Given f(t) = at^3 + t^2 - 2t + b.
Given g(t) = t^2 - 1.
= (t + 1)(t - 1).
(1)
Since, (t + 1) is a factor of f(t), therefore f(-1) = 0.
= > f(-1) = a(-1)^3 + (-1)^2 - 2(-1) + b
= > -a + 1 + 2 + b = 0
= > -a + 3 + b = 0
= > b - a = -3
= > -(a - b) = -3
= > a - b = 3 ----- (1)
(2)
since,(t - 1) is a factor of f(t), therefore f(1) = 0
= > f(1) = a(1)^3 + (1)^2 - 2(1) + b = 0
= a + 1 - 2 + b = 0
= a - 1 + b = 0
= a + b = 1 ------ (2)
On solving (1) & (2), we get
= > a - b = 3
a + b = 1
--------------
2a = 4
a = 2.
Substitute a = 2 in (1), we get
= > a - b = 3
= > 2 - b = 3
= > -b = 1
= > b = -1.
Therefore, the value of a = 2 and b = -1.
Hope this helps!...