Polynomials again Terminology: Monomial in x Example 3*a^2*x^3 Coefficient and degree of a monomial in x 3*a^2 degree 3 Polynomial in x x + a*x 2*x^2 coefficients and degree of a polynomial in x 1, a, 2 degree 2

Diagram showing relations between whole numbers, integers, rational numbers, irrational numbers, real numbers, complex numbers. You had to be there

Expressions are written in various ways depending on whose reading them. We use algebra to convert from form to another. ‘book’ form ‘calculator’ syntax

Three operations with polynomials Let f(x) = x^2 + 2*x + 1 = and g(x) = x + 2 Addition: f(x) + g(x) = x^2 + 2*x + 1 + x + 2 = x^2 + 3*x + 3 Multiplication: f(x) * g(x) = (x^2 + 2*x + 1)*( x + 2) = x^3 + 2*x^2 + x + 2*x^2 + 4*x + 2 = x^3 + 4*x^2 + 5*x + 2 Substitution: f(g(x)) = f(x+2) = (x+2)^2 + 2*(x+2) +1= x^2 + 4*x + 4 + 2*x + 4 + 1 = x^2 + 6*x + 9

Question: determine the degree of the sum, product, and substitution polynomia Degree (f(x) + g(x)) <= Max(degree(f),degree(g)) Degree( f(x)*g(x)) = degree(f) + degree(g) Degree(f(g(x)) = degree(f)*degree(g)

Binomial expansion = = = The product is the sum of all monomials that can be made by taking one term out of each factor. More generally The sum of all monomials of type where there are n terms, each taken from exactly one of the factors.

is the sum of all monomials where the total number of a’s and b’s is n If there are j a’s then there will be n – j b’s so taking commutativity into account such a term must be This says, for instance So to write down a standard form of we just need to know the coefficients. 1 1 1 1 2 1 1 3 3 1

Binomial Expansion (Pascal’s Triangle) 1 1 1 1 2 1 This says, for instance 1 3 3 1 1 4 6 4 1 1 __ __ __ __ _ Why does it work?

Application Problem: Flip a coin 6 times. How many ways to get a head exactly 2 times? What is the probability of getting exactly 2 heads in 6 flips? What is the coefficient of the x^2 term in the polynomial (1+x)*(1+x)*(1+x)*(1+x)*(1+x)*(1+x) Answers: 15, 15/64

Problem: Suppose a rectangle has width x + 3 ft. and height x ft. Then Area = x*(x+3) = x^2 + 3*x sq. ft. Perimeter = x + x + (x +3)+(x+3) = 4*x + 6 ft

Follow up question Suppose the perimeter is 100 ft. What is the area? Solve 4*x + 6 = 100 for x to get x. Then plug into area x^2 + 3*x x = 94/4 = 47/2 = 23.5 ft Area = 23.5^2 + 3*23.5 = 622.75 sq.ft.

Another followup Suppose the area is 150 square feet. What is the perimeter? We didn’t get to this one.