Calculate lim_(x rarr oo)(2x^(2) - 2x + 3)/(x^(2) + 4x + 4) .

Simplify (x^(2) + x-6)/(x^(2) + 4x +3)

If two polynomials 2x^(3) + ax^(2) + 4x -12 and x^(3) +x^(2) - 2x + a leave the same remainder when divided by ( x-3) , find the value of a. and also find the remainder.

If two polynomials 2x^(3) + ax^(2) + 4x-12 and x^(3) + x^(2)-2x+a leave the same remainder when divided by (x – 3) , find the value of a and also find the remainder.

Use the Factor Theorem to determine whether g (x) is factor of f(x) in each of the following cases: (i) f (x) = 5x ^(3) + x ^(2)-5x -1, g (x)=x +1 (ii) f (x) = x ^(3) + 3x ^(2) + 3x +1 , g(x) =x +1 (iii) f (x) =x ^(3) - 4x ^(2) +x + 6, g (x) =x -2 (iv) f (x) = 3x ^(3) + x^(2) - 20x + 12, g (x) = 3x -2 (v) f (x) = 4x ^(3) + 20 x ^(2) + 33 x +18, g (x) =2x +3

By remainder theorem, find the remainder when, p(x) is divided by g(x) where, p(x)=4x^(3)-12x^(2) + 14x-3, g(x)= 2x-1

Analyze the roots of the following equations: (i) 2x^(3) - 9x^(2) + 12x - (9//2) = 0 (ii) 2x^(3) - 9x^(2) + 12x - 3 = 0

Evaluate lim_(xrarrinfty)(4x^2 -2x +3 )/(2x^2 +3x -2)

Find (x^(2)+20x+36)/(x^(2)-3x-28)-(x^(2)+12x+4)/(x^(2)-3x-28)

Find int(1)/(x^(2)+4x+12)dx .