1. 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, degree 2

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

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

4. 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 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 *x
= x^2 + 6*x + 9

5. 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)

6. 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.

7. 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

8. Binomial Expansion (Pascal’s Triangle)1
This says, for instance
1 __ __ __ __ _
Why does it work?

9. 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

10. 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

11. 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 = sq.ft.

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