Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1
Example 1: Evaluate Numerical Expressions State which property of exponents you are using! Example 2: Use Scientific Notation in Real Life When dividing or multiplying scientific notation: 1. Divide or multiply the number parts and get a result. Round to 2 decimal places! 2. Use the Quotient of Powers Property or Product of Powers to deal with the exponent parts! 3. Check to make sure the number is written in scientific notation!!! Scientific Notation Reminder: The first number must be a number between 1 and 9.99! 10 X 10 0 = 1.0 X 10 1 (when you divide on the le, then you mulply on the right) 0.1 X 10 0 = 1.0 X 10 1 (when you mulply on the le, then you divide on the right) 2
Example 3: Simplify Expressions In simplified form, all exponents should be positive! State which properties of exponents you are using! Example 4: Standardized Test Practice Simplify. State which properties of exponents you are using! 3
Example 5: Compare Real Life Volumes Chapter 5 Lesson 2 Evaluate and Graph Polynomial Functions Vocabulary 4
Example 1: Identify Polynomial Functions How to determine if it is a polynomial function: 1. exponent must be a whole number (0,1,2...no negative numbers) 2. coefficients must be real numbers 3. terms must be monomials (no term can be divided by a variable) Example 2: Evaluate by Direct Substitution 5
Example 3: Evaluate by Synthetic Substitution Example 4: Standardized Test Practice 6
Example 5: Graph Polynomial Functions Use synthetic substitution to save time! State the domain and range. Domain: Range: Example 6: Solve a Multi Step Problem 7
Chapter 5 Lesson 3 Add, Subtract, and Multiply Polynomials Vocabulary Reminder: Like Terms Example 1: Add Polynomials 1. Combine Like Terms 8
Example 2: Subtract Polynomials 1. Distribute the Negative!!!! 2. Combine Like Terms Example 3: Multiply Polynomials 1. Distribute each term in the first polynomial to each term in the second polynomial 9
Example 4: Multiply Three Binomials 1. FOIL the first two binomials 2. Multiply the result by the last binomial Example 5: Use Special Product Patterns Reminders: a comes from the first term (always +) b comes from the second term (always +) 10
Example 6: Use Polynomial Models Chapter 5 Lesson 4 Factor and Solve Polynomial Equations Vocabulary 11
Example 1: Find a Common Monomial Factor Example 2: Factor the Sum or Difference of Two Cubes Reminders: a comes from the first term (always +) b comes from the second term (always +) 12
Example 3: Factor by Grouping How to factor by grouping: 1. Check if the polynomial has 4 terms! 2. Factor out a common monomial to one set of terms. Do the same with the other set. You must be left with a common binomial for this to work!!! At times, you may need to factor out a negative common monomial! 3. Rewrite the common binomial ONCE! Create the new binomial using the factored out pieces. 4. Make sure to check if there can be any other factoring completed! Example 4: Factor Polynomials in Quadratic Form 1. Always look for common factors first. 2. Check for quadratic form. *If the first exponent is twice the second exponent, then you can treat it similarly to the factoring we did in chapter 4. *You might also notice a special pattern that you could use (difference of squares, etc.). *Make sure you make the exponents on the variables correct! 3. Make sure to check if any other factoring can be completed! 13
Example 5: Standardized Test Practice How to Find the Solutions of Polynomials: 1. Factor first. Use all the methods we have discussed in this lesson! 2. Set each binomial equal to zero. Don't forget to put any variable pieces equal to zero as well! 3. Solve for all variables. 4. Check with your graphing calculator! Hint: the degree of the polynomial lets you know the number of zeroes that you are searching for (we will discuss the imaginary ones on the next example) Hint: the degree is 5 so we are searching for 5 zeroes! Example 6: Solve Polynomial Equations 14
Factoring Ideas to Think About: *Check for a common monomial! *If the polynomial has only 2 terms, check for perfect cubes or perfect squares! *If the polynomial has a degree of 2, think X Box! *If the polynomial has 4 terms, check if you can factor by grouping. *If the first term's exponent is double the second term's exponent, check if it is quadratic form and try X Box! *If, after factoring, any exponent is more than 1, always check again to see if you can factor again! 15