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Question

If the HCF of the polynomials p(x)=(x3)(2x2ax+2) and q(x)=(x2)(x2bx6) is h(x)=x25x+6, find the values of a and b.

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Solution

p(x)=(x3)(2x2ax+2) ........ (i)
and q(x)=(x2)(x2bx6) ........ (ii)
h(x)=x25x+6 is HCF of p(x) and q(x)
h(x)=x2+5x+6
=x23x2x+6
=x(x3)2(x3)
=(x3)(x2)
x=3,2 are the roots of p(x),q(x) and h(x)
p(x)=(x3)(2x2ax+2)=(x3)(x2)f(x), for some f(x)
(2x2ax+2)=(x2)f(x)
Take x=2
2(2)2a(2)+2=0
82a+2=0
a=5
Similarly, q(x)=(x3)(x2)g(x), for some factor g(x) of q(x)
(x2bx6)=(x3)g(x)
Take x=3
32b(3)6=0
33b=0
b=1
Hence, a=5 and b=1

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