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Question

Solve the following equations which have equal roots:
x63x5+6x33x23x+2=0.

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Solution

Give equation, x63x5+6x33x23x+2=0

Consider f(x)=x63x5+6x33x23x+2

f(x)=3(2x55x4+6x22x1)

Now, HCF of f(x) and f(x) is (x1)(x+1). Hence x=1 and x=1 are double roots of f(x)=0

f(x) can be factored as (x1)2(x+1)2(x23x+2)or(x1)3(x+1)2(x2)

Roots of the given equation are 1,1,1,1,1,2.


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