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.

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

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: degree of the polynomial = ______ leading coefficient = ___________ all coefficients: ________________ constant term = _____________

Example 2: 13 14 15 16 17 $ Birthday Expression Total 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: 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 Write a polynomial expression that represents the amount of money you will have at age 17. (In this case, x is the scale factor)

Example 3: Suppose a cube has a side of length (x+y). a. Label the cube to the right. Write the volume of the cube in expanded form. 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.