### Introduction to Polynomial Functions Investigating Polynomials

```Introduction to Polynomial Functions
Investigating Polynomials
Using each graph, fill in the important information about it, then answer the related questions at the end. Name by Degree: ______________ Name by Degree: ______________ Name by Degree: _____________ # of Zeros: _______ # of Zeros: _______ # of Zeros: _______ # of Max and Min: _______ # of Max and Min: _______ # of Max and Min: _______ y-­‐intercept: _______ y-­‐intercept: _______ y-­‐intercept: _______ Domain: _______ Domain: _______ Domain: _______ Range: _______ Range: _______ Range: _______ Name by Degree: ______________ Name by Degree: ______________ Name by Degree: _____________ # of Zeros: _______ # of Zeros: _______ # of Zeros: _______ # of Max and Min: _______ # of Max and Min: _______ # of Max and Min: _______ y-­‐intercept: _______ y-­‐intercept: _______ y-­‐intercept: _______ Domain: _______ Domain: _______ Domain: _______ Range: _______ Range: _______ Range: _______ Name by Degree: ______________ Name by Degree: ______________ Name by Degree: _____________ # of Zeros: _______ # of Zeros: _______ # of Zeros: _______ # of Max and Min: _______ # of Max and Min: _______ # of Max and Min: _______ y-­‐intercept: _______ y-­‐intercept: _______ y-­‐intercept: _______ Domain: _______ Domain: _______ Domain: _______ Range: _______ Range: _______ Range: _______ Name by Degree: _____________ Name by Degree: _____________ Name by Degree: _____________ # of Zeros: _______ # of Zeros: _______ # of Zeros: _______ # of Max and Min: _______ # of Max and Min: _______ # of Max and Min: _______ y-­‐intercept: _______ y-­‐intercept: _______ y-­‐intercept: _______ Domain: _______ Domain: _______ Domain: _______ Range: _______ Range: _______ Range: _______ Polynomial Investigation Questions
Q1: Compare the number of x-­‐intercepts of each graph and the greatest exponent found in its equation (degree). What is the relationship between degree and number of x-­‐intercepts? Q2: Compare the number of x-­‐intercepts of each graph and the greatest exponent found in its equation (degree). What is the relationship between degree and number of possible maximum or minimum points? Q3: Compare the y-­‐intercept values. How can the y-­‐intercept be determined from the polynomial function? Q4: Based on this observation, write a definition of the term “polynomial function.” Classifying Polynomials
Define polynomials: Research the meaning of the following words and fill in an equation and picture example that you find. I have completed one in each table for you. By Degree (highest power)
Name
Degree
3 Equation Examples
Picture Example
General Description
𝑦 = 3 Constant 0 𝑦 = −5 Horizontal line 𝑦 = 10 Linear Quadratic Cubic Quartic Quintic By Number of Terms
Name
Number of Terms
Monomial 3 Equation Examples
Binomial Trinomial Polynomial of ___ Terms 4+ General Description
𝑦 = 𝑥 ! − 3𝑥 ! + 2𝑥 ! − 7 𝑦 = 2𝑥 ! + 5𝑥 ! + 𝑥 ! + 4𝑥 − 6 𝑦 = 3𝑥 ! + 5𝑥 ! + 10𝑥 + 5 Any polynomial with 4 or more terms is just called a polynomial of that many terms. Name each of the following by both degree and number of terms. Standard Form: a polynomial that has terms arranged from highest degree to lowest degree. Arrange the following polynomials in standard form. Then, name each by both degree and number of terms. Identifying Polynomials and Zeros from Factored Form
• What is the degree of each polynomial? • Find all zeros. 1. 𝑓 𝑥 = (𝑥 + 4)(𝑥 − 3)(2𝑥 − 7) 2. 𝑓 𝑥 = 4𝑥(𝑥 − 7)! (𝑥 + 3) 3. 𝑓 𝑥 = (𝑥 − 3)(𝑥 ! + 2𝑥 + 1) 4. 𝑓 𝑥 = (𝑥 ! + 25)(𝑥 + 1)(2𝑥 − 5) 5. 𝑓 𝑥 = 𝑥 ! − 12𝑥 ! − 64 6. 𝑓 𝑥 = 𝑥 ! − 16 + Matching – match each graph with its function
1. _________
2. _________
3. _________
4. _________
5. _________
Extra Practice
7. Refer to the following polynomial to answer the questions.
Find all zeros.
. 7.) 𝑥 ! − 𝑥 ! − 72 = 0
8.) 𝑥 ! − 2𝑥 ! = 35𝑥
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