Download presentation

Presentation is loading. Please wait.

Published byTrinity Green Modified over 4 years ago

1
FACTORS OF INTEGERS AND POLYNOMIALS Section 4.1

2
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)

3
Integers and polynomials have many of the same rules. For example, if I add two integers, I must get an integer. ( 3 + - 5 = -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.

4
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

5
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).

6
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)

7
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.

8
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

9
Example 4: Let a (x) = x + 7 b(x) = x 2 – x – 56 and c(x) = 9x 2 + 63x 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 2 + 62x - 56 2(5x 2 + 31x – 28)(5x + 4) (x – 7) Factor of a Polynomial Sum Theorem

10
Homework Pages 226 – 227 1 – 10, 13, 14

Similar presentations

Presentation is loading. Please wait....

OK

Equivalence Relations

Equivalence Relations

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google