1. Introduction to Polynomial Functions

2. (6.1) Polynomial Functions
Monomial – a real number, variable, or product of the two.
Polynomial – a monomial or sum of monomials.
Degree – Exponent of the variable in a term.
Standard Form of a Polynomial – Degrees are in descending order. No two terms have the same degree.
P(x) = 2x x x
Cubic
Term
Quadratic
Term
Linear
Term
Constant
Term

3. Algebra 1 Review – Simplify and write in Standard Form
Ex) (2x2 – 5) + (4x2 + 9)
Ex) (4x2 – 6x + 2) – (5x – 7)
Ex) x(x + 3)2
Ex) (2x – 4)(2x + 6)(3x – 8)

4. YOU TRY: Simplify and write in Standard Form
Ex) (4x2 +7) + (3 – 5x2)
Ex) (8x2 + 4x – 8) – (6x2 – 3)
Ex) x(3x – 2)2
Ex) (2x – 5)(3x – 6)(2x + 1)

5. Name Using Number of Terms
Degree of a Polynomial – the largest degree of any term
Degree
Name Using Degree
Polynomial
Example
Number of Terms
Name Using Number of Terms
Constant 1
Monomial
Linear 2
Binomial
Quadratic 3
Cubic
Trinomial 4
Quartic 5
Quintic
Poly. of 4 Terms

7. (6.2) Polynomials and Linear Factors
What are the prime factors of 36?
What are the prime LINEAR factors of x2 + 4x - 12
Ex) Write the expression (x – 1)(x + 3)(x + 4) as a polynomial in standard form.
You Try: (x + 1)(x + 1)(x + 2)

8. Ex) Find the zeros of the function below and graph:
The x-intercepts of the graph of a function are called zeros because the value of the function is zero at each x-intercept.
Ex) Find the zeros of the function below and graph:
y = (x – 7)(x – 5)(x – 3)
Ex) Create our own…

9. We can reverse the process and write linear factors when you know the zeros.
FACTOR THEOREM
The expression (x – a) is a linear factor of a polynomial if and only if the value a is a zero of the related function.

10. Ex) Write a function in standard form with zeros at - 4, - 2, and 1.
Multiple Zero – a repeated zero.
Multiplicity – the number of times the zero occurs.
Ex) For each function, find any multiple zeros and state the multiplicity.
a) f(x) = (x - 2)(x+1)(x+1)2
b) y = x3 – 4x2 + 4x

11. Ex) Using factoring to find maximum values:
A metalworker wants to make an open box from a sheet of metal, by cutting equal squares from each corner as shown.
a) Write an expression for the length, width, and height of the open box.
b) Use your answer from part (a) to write a function for volume.
c) Graph the function. Find the maximum volume that can be contained by the box and the size of the square cut that produces this volume. (Relative Maximum) x 12 16