Question

Definition. The degree of a a polynomial is the exponent on the the highest power of x. Polynomial Degree 210 - 5.0 + 6 10 3.C - 1 13 Exercise 4. Scheinerman Exercise 35.12. Consider polynomials in x with rational coeffi- cients. a) Suppose p and q are polynomials. Write a careful definition of what it means for p to divide q (i.e. plq). Verify that (2.1 – 6(x3 – 3.x2 + 3x – 9) is true in your definition. b) Give an example of two polynomials p and q such that p+q but plq and qp. c) What is the relationship between polynomials that divide each other? d) Prove the following analogue of Theorem 35.1: Let a and b be polynomials such that b + 0. Then, there exist polynomials q and r so that a = qb+r with deg r < deg b e) In this generalized version of Theorem 35.1, are the polynomials q and r uniquely deter- mined by a and b?

Answer 1

Given: a) Suppose p and q are polynomials Plq meamy qkCX) PCM) Where KCHSE acres 23-34?+34-2 = x? (1-3)+304-3) = (x-3) (342+3) =< (2243) (264-3)) .. 2x-6 | 23-3x2+38-9 b) Let P-X-3, q= -(x-3) Then ptq and Plq and qlp c) Let Plq and alp q CH) - Kecu) PCV) i alp> PCX)= K2C4) 904 so, 90x) = Kicx) PCM - 2K1(x) Kz (A) q CH) .: 90x) to Kieu K₂ C) = 1 K, (x) & K₂ (1) boatte are constant Let kice) aka K2(X) = K2 K, K₂=1 - K₂= 1lk, d) a and bare polynomials such that bto By divison algorittim, we get - az abtr with deg & Ldeg b e) q and are conique a=q, 6 tri=q26th :.62,-92) 6 = 0, -r2 ~ deg of 2degb.. from we have 81-82=0 21282 9,2%