Provide a proof for the following statement: Let f(x) = anxn + an-1xn-1+ middot

Question

Provide a proof for the following statement: Let f(x) = anxn + an-1xn-1+ middot middot middot + a1x + a0 be a polynomial of degree n with integer coefficients. If p/q is rational root f(x), where p and q are integers with greatest common divisor 1, then p divides a0 and q divides an. Use the statement in Question (1) to prove the following Corollary: If z is un integer root of the polynomial f(x) = xn + middot middot middot + a1x + a0 of degree n with integer coefficients, then z divides a0. Apply the statement in Question (1) to prove by exhaustion the following proposition. There are no rational roots of the polynomial g(x) = 2x8 - x7 + 8x4 + x2 - 5.

Explanation / Answer

Q1)

for the given equation, product of roots = (-1)n.ao/an = (p/q)(x1.x2.x3......xn-1)

where x1,x2,x3..... are roots of f(x)=0

since p and q are coprime and ao/an is rational, so ao has a factor of p while an has a factor of q. PROVED

Q2) if z is an integer root then q=1 in p/q.

so, working similarly ao has a factor of z now. Proved

Q3)for g(x)=0

product of roots = -5/2

so if a rational root p/q exists then p is a factor of 5 and 2 is a factor of 2.

Factors of 5 : 1

Factors of 2 : 1

But p and q are coprime. Hence no such rational number exists.

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