divisibility of polynomials

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1/12/2015 Alg: HW 11 HW 11: HW assignments for Unit 13. The solution file must be in readable PDF format with the file size < 2M, and it must be submitted through BlackBoard. The BlackBoard does not know how to handle a file name containing special characters such as ”#”, ”-”, ”&” and so on (as well as an empty space ” ”). So, when you are submitting your HW through BlackBoard, please name your PDF file to say ”IwashitaHW11.pdf” (please substitute your LAST name in place of ”Iwashita”!). Note that I am refering to the name of the actual file; not the title you use to submit through BlackBoard. Remark: Before attempting to work on any of the problems below, you should ask yourself whether you know the definition of divisibility of polynomials or not. If not, look it up! You will go nowhere without knowing the precise statement of the definition of divisibility. Problem A. (5 pts) Let f (x), g(x), h(x) 2 Z[x] such that f (x) 6= 0. Prove that: If f (x) | g(x) and f (x) | [x2 · h(x)], then f (x) | [(2x + 1)g(x) + (2x5 + 4x3 )h(x)]. [Hint: Note that (2x5 + 4x3 )h(x) = (2x3 + 4x)x2 · h(x).] Problem B. (5 pts) Let f (x), g(x), h(x) 2 Z[x] such that f (x) 6= 0. Prove that: If f (x) | g(x) and f (x) | [3x5 g(x) h(x)], then f (x) | h(x). Remark: Before attempting to work on the Problem C below, you should ask yourself whether you know the definitions of divisibility of polynomials and the degree of a polynomial or not. If not, look it up! You will go nowhere without knowing the precise statement of the definitions. Problem C. (2 pts) Find two integral polynomials f (x), g(x) 2 Z[x] such that deg(f (x)) = 6 and deg(f (x) + g(x)) = 3. Problem D. (3 pts) Is there a pair of integral polynomials f (x), g(x) 2 Z[x] such that deg(f (x)) = 7 and deg(f (x) · g(x)) = 5? Give your reasons. 1
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