1.

For which values of a and b are zeroes of q(X)=x^(3)+2x+a also zeroes of the polynomial p(x)=x^(5)-x^(4)-4x^(3)+3x^(2)+3x+b? Which zeroes of p(x) are not zeroes of p(x)?

Answer»

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Solution :`:'` Zeroes of Q(x) are ALSO zeroes of p(x).
`:. ""` q(x) p(x) is a factor of p(x).
Now,`""` divide p(x) by q(x).

By factor THEOREM
r(x)=0
`implies(-a-1)x^(2)+(3+3a)x+(b-2a)=0`
`implies "" -a-1=0 "" and ""3+3a=0 "" and ""b-2a=0`
`implies "" a=-1"" and ""a=-1 "" and ""b=2a`
`implies "" a=-1 "" and ""b=-2""` Ans.
Now,the other zeroes are given by
`x^(2)-3x+2=0`
`implies "" x^(2)-2x-x+2=0`
`implies "" x(x-2)-1(x-2)=0`
`implies "" (x-2)(x-1)=0`
`implies "" x-2=0 "" or ""x-1=0`
`implies "" x=2 "" or ""x=1 ""` Ans.


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