For the polynomial `f(x) = a_0x^n + a_1x^(n-1) +... + a_(n-1)x + a_n`, find the limit as `x -> oo or x -> -oo`.

If x-r is a factor of the polynomial f(x)=a_0x^n+a_1x^(n-1)+.....+a_n repeated m times, (1ltxlt=n), then r is a root of f^(prime)(x)=0 repeated m times.True/False.

If x-r is a factor of the polynomial f(x)=a_0x^n+a_1x^(n-1)+.....+a_n repeated m times, (1ltxlt=n), then r is a root of f^(prime)(x)=0 repeated m times.

If x-r is a factor of the polynomial f(x)=a_0x^n+a_1x^(n-1)+.....+a_n repeated m times, (1ltxlt=n), then r is a root of f^(prime)(x)=0 repeated m times.True/False.

If the remainder of the polynomial a_0 + a_1x + a_2x^2 + .... + a_n x^n when divided by (x - 1) is 1, then which one of the following is correct ?

If the equation a_0x^n +a_1x^(n-1) +....a_(n-1)x=0 has a positive root alpha prove that the equation na_0x^(n-1) +(n-1)a_1 x^(n-2) + ---+a_(n-1) = 0 also has a positive root smaller than alpha

If n be a whole number and a_(0),a_(1),a_(2),..............,a_(n)(a_(n)ne0) are constants, then the polynomial p(x)=a_(n)x^(n)+a_(n-1)x^(n-1)+.............+a_(1)x+a_(0) will be a zero polynomial, when

a_(0)x^(n) + a_(1)x^(n-1) + a_(2)x^(n-2) + ……..a_(n)x^(n) is polynomial of degree ……………

let f(x)=lim_(n rarr oo)(x^(2n)-1)/(x^(2n)+1)