Warm-Up f(x) Copy the coordinate plane with the following information. X - X Simplify each expression. f(x) - 4) (x + 5) + (2x + 3) 5) (x + 9) – (4x + 6) 6) (-x2 – 2) – (x2 – 2)

An Intro to Polynomials Essential Questions: How can we identify, evaluate, add, and subtract polynomials? How can we classify polynomials, and describe their end behavior given the function?

Classification of a Polynomial by Degree Name Example n = 0 constant 3 n = 1 linear 5x + 4 n = 2 quadratic 2x2 + 3x - 2 n = 3 cubic 5x3 + 3x2 – x + 9 n = 4 quartic 3x4 – 2x3 + 8x2 – 6x + 5 n = 5 quintic -2x5 + 3x4 – x3 + 3x2 – 2x + 6

Classification of a Polynomial by Number of Terms Name Example monomial 3 binomial 5x + 4 trinomial 2x2 + 3x - 2 polynomial 5x3 + 3x2 – x + 9 binomial -2x5 + 3x2 – x5 - 3x2 + 6 Combine like terms

Example 1a Classify each polynomial by degree and by number of terms. a) 5x + 2x3 – 2x2 cubic trinomial b) x5 – 4x3 – x5 + 3x2 + 4x3 quadratic monomial

Description of a Polynomial’s Graph Example f(x)= 3 f(x)= 5x + 4 f(x)= 2x2 + 3x - 2 f(x)= 5x3 + 3x2 – x + 9 f(x)= 3x4 – 2x3 + 8x2 – 6x + 5 f(x)= -2x5 + 3x4 – x3 + 3x2 – 2x + 6

Example 1b Determine the polynomial’s shape and end behavior. a) f(x)= 5x + 2x3 – 2x2 b) f(x)= x5 – 4x3 – x5 + 3x2 + 4x3

Example 2 Add (5x2 + 3x + 4) + (3x2 + 5) = 8x2 + 3x + 9

Example 3 -3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1 5x5 - 3x3y3 - 5xy5 5x5 Add (-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1) + (5x5 – 3x3y3 – 5xy5) -3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1 5x5 - 3x3y3 - 5xy5 5x5 – 3x4y3 + 3x3y3 – 6x2 + 1

Example 4 Subtract. (2a4b + 5a3b2 – 4a2b3) – (4a4b + 2a3b2 – 4ab)

Example 5 If the cubic function C(x) = 3x3 – 15x + 15 gives the cost of manufacturing x units (in thousands) of a product, what is the cost to manufacture 10,000 units of the product? C(x) = 3x3 – 15x + 15 C(10) = 3(10)3 – 15(10) + 15 C(10) = 3000 – 150 + 15 C(10) = 2865 $2865

Graphs of Polynomial Functions Graph each function below. Function Degree # of U-turns in the graph y = x2 + x - 2 2 1 y = 3x3 – 12x + 4 3 2 y = -2x3 + 4x2 + x - 2 3 2 y = x4 + 5x3 + 5x2 – x - 6 4 3 y = x4 + 2x3 – 5x2 – 6x 4 3 Make a conjecture about the degree of a function and the # of “U-turns” in the graph.

Graphs of Polynomial Functions Graph each function below. Function Degree # of U-turns in the graph y = x3 3 y = x3 – 3x2 + 3x - 1 3 y = x4 4 1 Now make another conjecture about the degree of a function and the # of “U-turns” in the graph. The number of “U-turns” in a graph is less than or equal to one less than the degree of a polynomial.

Example 6 Graph each function. Describe its end behavior. a) P(x) = 2x3 - 1 b) Q(x) = -3x4 + 2

Homework