MATH 2Z03 Lecture 7: Math 2Z03 - 3.4 Undetermined Coefficients (Review)

Polynomials and Derivatives In this project you will explore an alternative polynomial. Th once (first, second,...). The drawback is that it takes a while to work through all the algebra and arrive at the desired results. You will be surprised by the simplicity of the method and will encounter topics related to this method in subsequent method for computing the derivatives of a e method is simple and relies on algebra alone. It also produces all derivatives at math classes. Definition: Polynomial A polynomial of a single real variable, x, is a function that can be written PCx)anxn +an-131-1a + ao where n is a nonnegative integer and ay are real numbers called the coefficients. If an 0, n is called the degree of the polynomial 1. State the Binomial Theorem. 2. Given the fourth degree polynomial P(x)42x3x5, Find P'(c), P"(c) P"(c), and P( (c) Given the fourth degree polynomial Px) +2x3 2+5, find P(u + c). Use the Binomial Theorem to expand P(u +c) and collect the terms to express P as a polynomial will real variable and coefficients dy(c) in terms of c 4. Find the relationship between the derivatives in part 2 and the coefficients di(c) in part3 Write a general formula for this relationship that would apply to any inth degree polynomial P. 5. For an nth degree polynomial P, write an expression for P(u + c) as we did in part 3; where u is the real variable and the coefficients are d,(c). Replace the u's with (x - c) and notice that you have an expression for P in terms of x again (P(x)- P(x-c+c) 6. Use the expression you found for an nm de gree polynomial P(x) in part 5 to show that your general formula from part 4 is true

1. Binomial Theorem: a t p" (e) = 12c2 + 12c-2 P" (c) 24c +12 p", (e) = 24