1.

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

Answer»


Solution :`:'` The zeroes of q(x) are also the zeroes fo p(x)
`:.` q(x) is a factor of p(x). `""` [`:'` DEGREE of q(x) is LESS than the degree of p(x)]
Polynomials
Now are division algorithm,

Here, SINCE q(x) is a factor of p(x). So, the remainder should be zero. i.e..,
`-(1+a)x^(2)+(3+3a)x+(b-2a)=0`
`implies "" 1+a=0, ""3+3a=0, "" b-2a=0`
`implies ""a=-1, "" a=-1, "" b=2a`
`implies "" a=-1, "" b=-2,`
`:.""q(x)=x^(3)+2x^(2)-1`
and`"" p(x)=x^(5)-x^(4)-4x^(3)+3X^(2)+3x-2`
Now, `"" p(x)=(x^(3)+2x^(2)-1)(x^(2)-3x+2)+0`
`=(x^(3)+2x^(2)-1)(x-1)(x-2)`
Other zeroes of p(x) are 1 and 2.


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