Find a polynomial function f(x) with integer coefficients and leading coefficient 1, such that f(x) has x= 5 84 as one of its roots.

Answer 1

for some reason when people mean 1/2 it comes out as 1 2 So maybe you meant x= 5/84, but it came out as 5 84?

if so then x=5/84 is a zero or root and

x-5/84 is a factor of the polynomial

x-5/84 = 0

84x -5 = 0

84x-5 is a factor

y = 84x-5, but that has leading coefficient 84 instead of 1

use other factors

y = f(x) = x(84x-5) = 84x^2-5x, but again 84 is the leading coefficient, so try

y = ax(84x-5) where 84a = 1, then a = 1/84

y = (1/84)x(84x-5) = x^2 -5x/84 but now the x term has a non integer coefficient. it's unavoidable

so maybe you really did mean 584 is the root or zero

then

f(x) = x^2 - 584x is a polynomial with leading coefficeint = 1 and all integer coefficients. There's an infinite number of polynomials that then meet the requirements of leading 1 and all integer coefficients