Question

5. Prove the Rational Roots Theorem: Let p(x)=ataiェ+ +anz" be a polynomial with integer coefficients (that is, each aj is an integer). If t rls (oherer and s are nonzero integers and t is written in lowest terms, that is, gcd(Irl'ls!) = 1) is a non-zero Tational root orp(r), that is, if tメ0 and p(t) 0, then rao and slan. (Hint: Plug in t a t in the polynomial equation p(t) - o. Clear the fractions, then use a combination of the resultsf Problems 3 and 4.)

Answer 1

Let P (x) = a_0 + a_1 a + ... + a_nx^n be a polynomial with integer coefficients Let t = r/s with gcd (r, s) = 1 is root of P (x) then P (t) = 0 implies P (r/s) = 0 a_0 + a_1 (r/s) + a_2 (r/s)^2 + ... + a_n (r/s)^n = 0 a_0 + a_1 r/s + a_2 r^2/s^2 + a_2 r^2/s^2 + ... + a_nr^n/s^n = 0 a_0 s^n + a_1 r s^n - 1 + a_2 r^2 s^n - 2 + ... + a_nr^n = 0/s^na_0 s^n + a_1 r s^n - 1 + a_2 r^2 s^n - 2 + ... + a_nr^n = 0_1 a_1 r s^n - 1 + a_2 r^2 s^n - 2 + ... + a_nr^n = -a_0 s^n clearly since r divides each term on LHS r must divide RHS implies r/a_0 s^n since gcd (r, s) = 1 implies gcd (r, s^n) = 1 for