Polynomial Degree and Finite Differences Objective: To define polynomials expressions and perform polynomial operations

Definitions:  Polynomial Expression: The sum of terms containing the same variable raised to different powers  Polynomial Function:  A function where all exponents are whole numbers and the coefficients are all real numbers  Written in the form a n x n + a n-1 x n-1 + … + a 1 x 1 + a 0

Examples:  Are each of the following polynomial functions? 1. f(x) = -5x 5 + 3x 4 – 2x + 8Yes 2. f(x) = 5 – 2xYes 3. f(x) = -8 Yes 4. f(x) = x x No 5. f(x) = 6x 2 – 5x -1 No 6. No

Definitions:  Terms – the number of monomials that make up the polynomial  Constant – the number without a variable  Polynomials are named by their terms:  Monomial: a polynomial that has one term  Binomial: a polynomial that has two terms  Trinomial: a polynomial that has three terms  Polynomial: if there are more than three terms

Definitions:  Degree of a monomial – the sum of the exponents of the variables  Degree of a polynomial – the degree of the term that has the highest degree  Leading coefficient – the coefficient of the first term when the polynomial is in standard form  Standard form – the degrees of the terms are written in descending order from left to right  Even or Odd function – determined by the degree of the function

Example:  f(x) = 2 – x + 5x 4 – 3x 2 + 2x 3  Standard form:  f(x) = 5x 4 + 2x 3 – 3x 2 – x + 2  Leading coefficient: 5555  Even or odd:  Even  Degree of each monomial:  4, 3, 2, 1, 0  Degree of the polynomial: 4444

Practice:  Identify the degree of each polynomial:  3x 4 – 2x 3 + 3x 2 – x + 7 4444  x 5 – 1 5555  0.2x – 1.5x x 3 3333  250 – 16x x 2222  x + x 2 – x 3 + x 4 – x 5 5555  5x 2 – 6x 5 + 2x 6 – 3x 6666

Practice:  Determine which of the expressions are polynomials. For each polynomial, state its degree and write it in standard form. If it is not a polynomial, explain why not.  1 + x 2 – x 3 polynomial?yesno  Standard form:-x 3 + x  0.2x x x 2 polynomial:yesno  Standard form:0.5x x x 2

Practice:  polynomial?yesno  can’t have a variable in a denominator  25polynomial?yesno  Standard form:25  polynomial?yesno  Standard form:  polynomial?yes no  Can’t have a variable inside a radical

Polynomial Operations  To simplify polynomials:  Combine like terms by adding or subtracting their coefficients  To add polynomials:  Combine like terms by adding or subtracting their coefficients  To subtract polynomials:  Remember that subtracting is the same as adding the opposite, so change the sign on the coefficient of each term being subtracted  To multiply polynomials:  Use the distributive property then combine like terms

Practice Problems:  (-x 2 + 2x – 1) + (6x 2 – x + 5)  5x 2 + x + 4  (2x 2 + 4x + 3) + (3x 2 – 9)  5x 2 + 4x – 6  (3x 2 – 2x + 6) + (-x 2 – 3)  2x 2 – 2x + 3  (7x 2 + x – 8) – (6x 2 – 3x + 5)  7x 2 + x – 8 – 6x 2 + 3x – 5  x 2 + 4x - 13

Practice Problems  (5x 3 – 5x 2 + 2x – 3) – (-x 2 + 4x + 3)  5x 3 – 5x 2 + 2x – 3 + x 2 – 4x – 3  5x 3 – 4x 2 – 2x – 6  x(7x 2 – 2x + 1)  7x 3 – 2x 2 + x  (x 2 + 3)(x 2 + 2x – 4)  x 4 + 2x 3 – 4x 2 + 3x 2 + 6x – 12  x 4 + 2x 3 –x 2 + 6x - 12

Practice Problems  (x + 3)(x – 1)(x + 5)  (x 2 – x + 3x – 3)(x + 5)  (x 2 + 2x – 3)(x + 5)  x 3 + 5x 2 + 2x x – 3x – 15  x 3 + 7x 2 + 7x - 15