If `f(x)` is a polynomial such that `f(a)f(b)<0,` then what is the number of zeros lying between `a` and ` b` ?`

If a and b are two distinct real roots of the polynomial f(x) such that a < b , then there exists a real number c lying between a and b , such that

Statement-1: The equation (pi^(e))/(x-e)+(e^(pi))/(x-pi)+(pi^(pi)+e^(e))/(x-pi-e) = 0 has real roots. Statement-2: If f(x) is a polynomial and a, b are two real numbers such that f(a) f(b) lt 0 , then f(x) = 0 has an odd number of real roots between a and b.

Let a and b be two distinct roots of a polynomial equation f(x) =0 Then there exist at least one root lying between a and b of the polynomial equation

Let f(x) be a non - constant polynomial such that f(a)=f(b)=f(c)=2. Then the minimum number of roots of the equation f''(x)=0" in "x in (a, c) is/are

If a quadratic polynomial f(x) is factorizable into linear distinct factors, then what is the total number of real and distinct zeros of f(x) ?

Assertion (A): Equation (x-a)(x-b)-2=0, altb has one root less than a and other root greater than b. , Reason (R): A polynomial equation f\'(x)=0 has even number of roots between a and b if f(a) and f(b) have opposite signs. .

Assertion (A): Equation (x-p)(x-q)-r=0 where p,q,r epsilon R and 0ltpltqltr has roots in (p,q) , Reason(R): A polynomial equation f(x)=0 has odd number of roots between a and b (altb) if f(a) and f(b) have opposite signs

Let f(x) be a polynomial of degree 5 such that f(|x|)=0 has 8 real distinct , Then number of real roots of f(x)=0 is ________.

Statement-1: If a, b, c, A, B, C are real numbers such that a lt b lt c , then f(x) = (x-a)(x-b)(x-c) -A^(2)(x-a)-B^(2)(x-b)-C^(2)(x-c) has exactly one real root. Statement-2: If f(x) is a real polynomical and x_(1), x_(2) in R such that f(x_(1)) f(x_(2)) lt 0 , then f(x) has at least one real root between x_(1) and x_(2)

If f(x) is a polynomial of degree <3, then, f(x)/((x−a)(x−b)(x−c))​ is equals to