Document Sample

3.2 Polynomials-- Properties of Division Leading to Synthetic Division Part 1 Some serious relationships Start with a POD Using long division, divide x4 - 16 by x2 + 3x + 1 By hand. On CAS. Start with a POD Using long division, divide x4 - 16 by x2 + 3x + 1 Quotient: x2 - 3x + 8 Remainder: -21x - 24 The POD leads to The Division Algorithm for Polynomials: f(x) = d(x) · q(x) + r(x) where d, q, and r are polynomials, d is the divisor, q is the quotient, r is the remainder, and r(x) has a smaller degree than d(x). There’s a pattern– what is it? Find the remainders with LINEAR divisors: x3 2 x 2 4 x 8 x 3 x3 2 x 2 4 x 8 x 3 Compare that to f(3)– the top polynomial. 2 x 4 3x 3 x 2 x 9 x 4 2 x 4 3x 3 x 2 x 9 x4 Compare that to f(-4). The Relationship The Division Algorithm for Polynomials: f(x) = d(x) · q(x) + r(x) What happens when the remainder = 0? The Factor Theorem: If the remainder = 0, then f(c) = 0. If f(c) = 0, then c is a root/ solution/ zero of f(x) and (x - c) is a factor of f(x). The Proof Start with: f(x) = d(x) · q(x) + r(x) What if d(x) = x - c ? (This is a linear divisor.) f(x) = (x - c)q(x) + r(x) r(x) must be of a degree less than (x - c), so r(x) must be a constant, k Let’s see what happens when we plug c in for x to find f(c): f(c) = (c - c)q(c) + k = 0·q(c) + k In other words, f(c) is the remainder k. The Remainder Theorem: If polynomial f(x) is divided by x - c, the constant remainder r(x) is f(c). Use it 1. If f(x) = x3 - 3x2 + x + 5, use the Remainder Theorem to find f(-4). 2. Show that (x - 2) is a factor of f(x) = x3 - 4x2 + 3x + 2. 3. Find a polynomial f(x) that has zeros at x = 2, -1, and 3. (How many could you find? How many that are 3rd degree? Use it 1. If f(x) = x3 - 3x2 + x + 5, use the Remainder Theorem to find f(-4). Divide f(x) by (x+4). 2. Show that (x - 2) is a factor of f(x) = x3 - 4x2 + 3x + 2. Show that f(2) = 0. 3. Find a polynomial f(x) that has zeros at x = 2, -1, and 3. f(x) = (x-2)(x+1)(x-3) (How many could you find? How many that are 3rd degree? The answer to both of these is “infinite.” Consider f(x) = (x- 2)(x+1)(x-3)2 to illustrate the first answer, and f(x) = 3(x- 2)(x+1)(x-3) to illustrate the second answer.) Part 2 Let’s make it easier. Dividing polynomials Use long division (not the easy part): x 3 2x 4 5x 3 2x 8 Dividing polynomials Use long division: 2x 3 x 2 3x 11 x 3 2x 4 5x 3 2x 8 With a remainder of 25. What is f(-3)? Dividing polynomials Now, use synthetic division: 2x 3 x 2 3x 11 becomes -3 |2 5 0 -2 -8 x 3 2x 4 5x 3 2x 8 |____________ What is the remainder? What is the quotient? Use this to find zeros of polynomials What polynomials are represented below? What is the remainder? The quotient? What is the full factorization over the set of integers? What does it all mean? -11 | 1 8 -29 44 |____________ Use this to find zeros of polynomials What polynomials are represented below? What is the remainder? The quotient? 0 x2 - 3x + 4 What does it all mean? For f(x) = x3 + 8x2 - 29x +44 -11 is a zero/ solution/ root (x + 11) is a factor of f(x) What are the integer factors of f(x)? (x + 11)(x2 - 3x + 4) Use this to find other information about polynomials For f(x) = x3 + 8x2 - 29x +44 use synthetic division to find f(2). Use this to find other information about polynomials For f(x) = 2x4 + 5x3 - 3k, What is k if f(2) = 39? Summary (using different letters) If f(a) = b, then 1. b is the remainder of f(x)/(x - a). 2. (a, b) is on the graph of f(x). 3. The value of f(x) at a is b. If b = 0 also, then 1. (x - a) is a factor of f(x). Note: linear factor! 2. a is a zero of f(x). 3. (a, 0) is on the graph of f(x) 4. a is an intercept of f(x). 5. a is a solution of f(x) = 0. Use it one more time 1. Give a factor of f(x) = x3 - 5x2 - 6x if (6, 0) is on the graph. What is another factor? 2. If (-4, k) is on the graph of x3 - 2x + 2, what is k? How many ways could you corroborate your answer?

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 3 |

posted: | 5/30/2012 |

language: | English |

pages: | 20 |

OTHER DOCS BY ert554898

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.