An algebra problem by Pronay Biswas
Algebra
Level
3
How many integer roots the equation has?
No integer root
4
2
1
3
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1 solution
no integer root will exist.
let f(x) is a nth degree polynomial if f(x) and f(1) be both odd then f(x) cannot have an integer root.
GENERALLY>>
theorem= let, n be an integer root of a nth degree polynomial f(x) iff f(0), f(1).......,f(n-1) are divisible by n then n cannot be an integer root. Proof=> let, c be an integer solution of f(x)=0
let, c=nk+j (j=0,1,2,3,......n-1)
=>c-j = nk
=>c=j (mod n)
=>f(c)= f(j) (mod n)
=> f(j)=o (mod n)
this implies n| f(j), which is a contradiction :)