Do Now

• Find an expression representing the surface area and volume of a box with one edge unknown, a second edge one unit longer that the first edge, and the third edge one unit shorter than the first edge.

Chapter 9: Polynomial Functions

Lesson 1: Polynomial Models

Mrs. Parziale

Vocabulary

polynomial: a polynomial in (x) is an expression of the form

where (n) is a nonnegative integer and

Ex.

1 21 2 1 0....n n n

n n na x a x a x a x a

0na

4 3 23 7 5 10x x x x

Degree of a Polynomial

• For a polynomial in a single variable - the value of the highest exponent.

• For a polynomial of multiple variables - the largest sum of the exponents of the variables in any term of the polynomial.

More Vocab

• coefficient: numbers in front of the variables. (an, an-1, an-2, ...)

• leading coefficient: an (number in front of the variable with the largest exponent)

• standard form (for a polynomial function in 1 variable): the polynomial is written in descending order of exponents

1 21 2 1 0....n n n

n n na x a x a x a x a

Naming Polynomials

• Based on number of terms: –monomial - polynomial with one term –binomial – polynomial with two terms – trinomial – polynomial with three terms

Example 1:

• Given the polynomial

5x7 + 6x5 – 4x4 + 7x3 + 3x2 – 2 Identify:

(a) degree of the polynomial = ______

(b) leading coefficient = ___________

(c) all coefficients: ________________

(d) constant term = _____________

Example 2: • Suppose that each year since your 13th birthday, your parents have given you

money to save for college. The first year they give you $500, the next year 700, etc. (Each year they give you $200 more than the last year.) You invest it in an account paying 5% interest compounded annually. How much money will you have when you turn 17?

Birthday Expression Total

13 $

14 $

15 $

16 $

17 $

Example 2:

• Write a polynomial expression that represents the amount of money you will have at age 17. (In this case, x is the scale factor)

Birthday Expression Total

13 500 $500

14 500(1.05) + 700 $1225

15 500(1.05)2 + 700(1.05) + 900 $ 2186.25

16 500(1.05)3 + 700(1.05)2 + 900(1.05) + 1100 $ 3395.56

17 500(1.05)4 +700(1.05)3 +900(1.05)2 +1100(1.05) +1300

$ 4865.34

Example 3:

• Suppose a cube has a side of length (x+y).

a. Label the cube to the right.b. Write the volume of the cube

in expanded form. c. What is the degree of the

polynomial in Example 3?

Closure: Example 4

A company wants to construct a box. They take a 50” by 75” piece of cardboard, cut out a square of length (p) from each corner, and fold the box (with no lid).

a. Label the figure at the right.b. Express the volume of the box as a

polynomial in standard form.