FACTORS OF INTEGERS AND POLYNOMIALS Section 4.1

5x 4 + 3x 3 + 9x 8 – 15x 5 + 2x 14 Polynomial Coefficients Leading Coefficient Leading Term Degree of the polynomial d(x) is a factor of n(x) iff there exists a polynomial q(x) such that n(x) = q(x) * d(x)

Integers and polynomials have many of the same rules. For example, if I add two integers, I must get an integer. ( = -2) this applies to polynomials as well. The set of both integers and polynomials is said to be closed under addition, subtraction, multiplication, and division.

Theorems Transitive Property of Integer/Polynomial Divisibility: a is a factor of b b is a factor of c then a is a factor of c 3 is a factor of 6 6 is a factor of 18 then3 is a factor of 18

Factor of an Integer Sum: If a is a factor of b, a is a factor of c then a is a factor of b + c. Factor of a Polynomial Sum: If a(x) is a factor of b(x), a(x) is a factor of c(x), then a(x) is a factor of (b + c)(x).

Example 1 For all integers x and y, a. Is -3 a factor of 54y – 12x + 13? b. Is 6xy 2 a multiple of xy? No, because I cannot take out -3 from each Yes! Because I can factor xy(6y)

Example 2: If a, b, c, and d are in the set of integers, then a 5 (b + c)d 3 – d 2 c is also an integer Yes, closed under add, sub, and mult.

Example 3: Is the converse of the factor of an Integer Sum Theorem true? That is, is it true that for all integers a,b, and c, if a is a factor of b + c, then a is a factor of b and a factor of c? If true, prove the statement. If false, provide a counterexample. a = 3b + c = 15 b = 8 but 3 is not a factor of 8 c = 73 is not a factor of 7

Example 4: Let a (x) = x + 7 b(x) = x 2 – x – 56 and c(x) = 9x x Show a(x) is a factor of b(x) and c(x) Show b(x) + c(x) can be expressed as a(x) * some polynomial What theorem is illustrated by parts a and b? b(x) = a(x) * (x – 8)c(x) = a(x) * 9x 10x x (5x x – 28)(5x + 4) (x – 7) Factor of a Polynomial Sum Theorem

Homework Pages 226 – – 10, 13, 14