If t3– 2t2– pt + q is divided by t2– 2t – 3, the remainder is t – 6 then,find the pair of linear equations in p and q.
Answers
Given : t³– 2t²– pt + q is divided by t²– 2t – 3, the remainder is t – 6
To Find : values p and q.
Solution:
Dividend = Divisor x Quotient + remainder
f(x) = g(x)q(x) + r(x)
Dividend = t³– 2t²– pt + q => degree 3
Divisor t²– 2t – 3 => degree 2
Hence degree of Quotient = 3 - 2 = 1
Let say Quotient is at + b
t³– 2t²– pt + q = (t²– 2t – 3)(at + b) + t - 6
=> t³– 2t²– (p+1)t + (q+ 6) = (t²– 2t – 3)(at + b)
Solving RHS
= (t²– 2t – 3)(at + b)
= at³ + (b - 2a)t² - (2b + 3a)t -3b
Equating with LHS
t³– 2t²– (p+1)t + (q+ 6)
a = 1
b - 2a = -2 => b - 2 = -2 => b = 0
-3b = q + 6 => 0 = q + 6 => q = - 6
2b + 3a = p + 1
=> 3 = p + 1
=> p = 2
p = 2 and q = - 6
t
t²– 2t – 3 ) t³– 2t²– 2t - 6 (
t³– 2t²– 3t
____________
t - 6
Hence verified
p = 2 and q = - 6
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