1.

Find k so that x^(2)+2x+k is a factor of 2x^(4)+x^(3)-14x^(2)+5x6. Also find all zeroes of two polynomials.

Answer»

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Solution :LET `"" p(x)=2x^(4)+x^(3)-14X^(2)+5x+6`
and `"" g(x)=x^(2)+2x+k`

But remainder =0
`IMPLIES""(21+7k)x+(2k^(2)+8k+6)=0`
`implies ""21+7k=0 ""and ""2k^(2)+8k+=0`
`implies "" 7k=-21 "" and "" k^(2)+4k+3=0`
`implies "" k=-3"" and ""(k+3)(k+1)=0`
`implies "" k=-3 ""and ""k=-3 or k=-1`
`:. "" k=-3`
Now, `q(x)=2x^(2)-3x-8-2k=2x^(2)-3x-8+6=2x^(2)-3x-2`
`=2x^(2)-4x+x-2=2x(x-2)+1(x-2)=(x-2)(2x+1)`
Its zeroes are given by x-2=0 and 2x+1=0
`implies ""x=2 andx=-(1)/(2)`
and `"" g(x)=x^(2)+2x+k=x^(2)+2x-3=x^(2)+3x-x-3`
`=x(x+3)-1(x+3)=(x+3)(x-1)`
Its zeroes are given by x+3=0 and x-1=0
`implies"" `x=-3 and x=1
`:.` Zeroes of g(x) are-3and 1 and zeroes of p(x) are -3, 1, 2 and `-(1)/(2) ""` Ans.


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