If ` x^(2) + ax + b ` is an integer for every integer x , then :

For some integer q , every odd integer is of the form

For some integer m , every even integer is of the form

Let f(x)=A x^2+B x+c ,w h e r eA ,B ,C are real numbers. Prove that if f(x) is an integer whenever x is an integer, then the numbers 2A ,A+B ,a n dC are all integer. Conversely, prove that if the number 2A ,A+B ,a n dC are all integers, then f(x) is an integer whenever x is integer.

If F(x)=Ax^(2)+Bx+C,and f(x) is integer when x is integer then

Let f(x)=x^(2)+ax+b be a quadratic polynomial in which a and b are integers. If for a given integer n, f(n) f(n+1)=f(m) for some integer m, then the value of m is

Let a, b be integers such that all the roots of the equation ( x^2 + ax + 20)(x^2 + 17x + b) = 0 are negative integers. What is the smallest possible value of a + b ?

In each of the following state if the statement is true (T) or false (F): (i) The quotient of two integers is always an integer. (ii) Every integer is a rational number. (iii) Every rational number is an integer. (iv) Every fraction is a rational number. (v) Every rational number is a fraction. (vi) If a/b is a rational number and m any integer, then a/b=(axxm)/(bxxm) (vii) Two rational numbers with different numerators cannot be equal. (viii) 8 can be written as a rational number with any integer as denominator. (ix) 8 can be written as a rational number with any integer as numerator. (x) 2/3 is equal to 4/6dot

Let f (x) =[x] and g (x) =0 when x is an integer and g (x) =x ^(2) when x is not an integer ([] is ghe greatest integer function) then:

(x+1) is a factor of x^n+1 only if (a)n is an odd integer (b) n is an even integer (c )n is a negative integer (d)n is a positive integer

When (2x - 3)^(2) is written in the form ax^(2) + bx + c where a, b and c are integers, a + b + c = ?