1. Polynomial Functions

2. Polynomial is an expression that is either a real number, a variable, or a product of real numbers and variables with whole number exponents. Standard Form Example: x 4 + 2x 3 – 3x 2 + 5x + 2 When we write a polynomial we follow the convention that says we write the terms in order of descending exponents, from left to right. Polynomials can be classified by their degree or number of terms.

3. Polynomial Functions View the link below to learn more about polynomials and their properties. Polynomial Functions

4. FTA Examples 1) 4x + 2 Degree = 1 (highest exponent) so the number of solutions is 1. 2) x 2 + 3x + 2 Degree = 2 so the number of solutions is 2. 3) 4x 3 + 3x 2 + 2x + 1 Degree = 3 so the number of solutions is 3. FTA is short for the Fundamental Theorem of Algebra The FTA states that the number of solutions to a polynomial equation is equal to the degree of the polynomial.

5. Zeros or Roots of a Function If a polynomial is in factored form, you can use the zero product property to find values that will make the polynomial equal zero! These values are called roots or zeros of the function…also known as the x-intercepts of the graph.

6. Example 1 Solve; x 2 -4x = 5 Set the equation equal to zero. x 2 - 4x – 5 = 0 Factor the left side of the equation(x - 5)(x + 1) = 0 Use the Zero Product Property If I multiply the two expressions on the left and product is equal to zero, one of the two must be equal to zero. Set each linear factor equal to zero. (x - 5)= 0 or (x + 1) = 0 Solve each equation x - 5 = 0 or x + 1 = 0 x = 5 x = -1

7. Example 1 Continued Multiplicity Let’s look at how we solved for x. (x – 5)(x + 1) = 0 Multiplicity is how often a certain root is part of the factoring. Notice that (x – 5)(x + 1) = 0 only occurred once so the multiplicity for (x – 5) and (x + 1) is 1.

8. Example 1 Continued Graph Let’s graph x 2 – 4x – 5 = 0 First we need to find the vertex. x = -b/2a x = -(-4)/2(1) = 4/2 = 2 y = (2) 2 – 4(2) – 5 = -9 Vertex = (2, -9) Then we graph can graph the x-intercepts (5, 0) and (-1, 0). Remember that you can graph polynomials In the graphing calculator.

9. Example 1 Cont. End Behavior The End Behavior is determined by how the function “behaves” as you move to the left or right. Both the left and right rise as you continue to graph the polynomial.

10. Let’s Practice PolynomialClassification by Degree Classification by Terms ZerosMultiplicity of Zeros End Behavior x 2 + 7x + 10 4x + 10 3x 3 – 12x Remember that you can graph these polynomials into your graphing calculator.

11. Let’s Practice - Solutions PolynomialClassification by Degree Classification by Terms ZerosMultiplicity of Zeros End Behavior x 2 + 7x + 102Trinomialx = 2 x = 5 1111 Left – Rises Right – Rises 5x + 101Binomialx = 21Left – Falls Right – Rises 3x 3 – 12x3Binomialx = 0 x = 2 x = -2 111111 Left – Falls Right – Rises