Question

Please prove the theorems, thank you

6.1 Theorem. Let anx+an-1- +ag he a polynomial of degree n0 with integer coefficients and assume an0. Then an integer r is a Poot of (x) if and only if there exists a polynomlal g(x) of degree n - with integer coeficients such that f(x) (x)g(x). This next theorem is very similar to the one above, but in this case (xr)g(x) is not quite equal to f(x), but is the same except for the constant term of f(x) and the constant term of (x r)g(x) . Thosc constant terms are not the same, but are congruent using an appropriate modulus. 6.2 Theorem. Let f(x) an" +a 1 1 + . + ag be a polynomial of degree n0 with integer coefficients and an0. Let p be a prime number and r an integer Then, if f(r)0 (mod p), there exists a polynomial g(x) of degree n 1 such that (r)g(x)anx" + ap-1x" where aobo(mod p) The final theorem of this section is a generalization of the Fundamental Theuren of Algebra in the setting of polynomials modulo a prime.

Answer 1

n- 6.1 f(2)=an a ,an #0 t an- with degree 1n polynomial a n 0 First inteqer 'r that uppore the tS a of fin) Then, by factor theo rem, f() 0 Now, it divide t(n) by (x- r) we the then theorem, remain der f(r) remaind to be OCCLLYA f(r)= 0 But have We we remainder it there So, no fx) dlivide by divimble fx) So, So, f)=-)3 () polynomial g() for 8ome degree n' f() is Now, mnce i of degree 10 and (n-1) x) iA ofdagree Thus, there polynomial g) exists a n-1)with integer degree coetficients [nin ce f(n) amd with integer coefficients Auch that flx) (x-) g(x) -r) are

Conversely uppose that, there exists 90x) of degree (n-) polynamial a with coetficient s Ruch that, integer where fa) in teger Aay We f fin) if a flo)0 ta) Here, 1 fr) rost of fin) Hence This the comp letes (Proved)

6-2 f(n) =an an- a poly nomial In of degree n> wTth integer efficienta co-e integer amd am prime divide We (n-) th en there palynomial fln) dagree and in ot degree Cn-) 0 degree n-1) &uch that ne get the tm) (x-Y)9(n) remaindeY Since remaindler (-r) theorem de gree 1 murt constant be ay A So, fln) -n -r) 9() t A 7 8aid t0 prynoials two wre be equal f dagree coeffieients hawe they Samme amd the have anme terms reapective eleonly f(n) amd etficienta 6 the all Ranme arre conktamt term. Except the

So, Let (-GAan t an- 1 Then have, from We 1 bo t A bo (mod p) = lo0t A So , a rod fr A 0 bDtA o o (mod p) So, Thus, n-) 9(7) ニ ana t an-i (n-r) ao bo (mod t) where, (Proved)