Bell Ringer 10 3 18 1 What are

Bell Ringer 10 -3 -18 § 1. What are the Properties of Exponents? § 2. How do we convert between exponential and radical form? § 3. How do we add, subtract, and multiply polynomials?

Properties of Exponents & Performing Operations with Polynomials (Adding, Subtracting, Multiplying, & Dividing by a Monomial) Wednesday, October 3, 2018

How to write exponents as radicals

What are Polynomials? §Mono- is a prefix meaning “one” §Bi- is a prefix meaning “two” §Tri- is a prefix meaning “three” §Poly- is a prefix meaning “many” §-nomial is a suffix meaning “terms” §Polynomial = Many Terms

How do we work with Polynomials? § We can perform operations with polynomials (addition, subtraction, multiplication, and division). § When we write polynomials, the standard is to write in descending order of the exponents. § The degree of a polynomial is the highest exponent or sum of the exponents if there are multiple variables in the same term.

Match the polynomial and its degree… § 1. – 3 § 2. a 3 – 2 a 2 § 3. 3 – 6 n 5 – 8 n 4 § 4. -10 x 4 y 3 + 6 y 3 + 4 x 4 y 4 5 p 2 § A. trinomial degree 8 § B. binomial degree 2 § C. trinomial degree 5 § D. binomial degree 3

Adding Polynomials is really… Combining Like Terms

Subtracting Polynomials is really… Adding the Opposite.

Examples: Adding Polynomials § (5 p 2 -3) + (2 p 2 – 3 p 3) § Identify like terms and combine them. 5 p 2 + 2 p 2 = 7 p 2 § Write in descending order of the exponents. § -3 p 3 + 7 p 2 -3 Subtracting Polynomials § (a 3 – 2 a 2) – (3 a 2 – 4 a 3) § Add the opposite, so… § (a 3 – 2 a 2) + (-3 a 2 + 4 a 3) § Combine like terms… a 3 + 4 a 3 = 5 a 3 -2 a 2 + -3 a 2 = -5 a 2 § 5 a 3 – 5 a 2

Note: It can be easier if you put the exponents in order first. §(-7 x 5 + 14 – 2 x) + (10 x 4 + 7 x + 5 x 5) §Reorder §-7 x 5 + 5 x 5 +10 x 4 – 2 x + 7 x + 14 §Combine like terms §-2 x 5 + 10 x 4 + 5 x + 14

Remember, when subtracting, add the opposite, then reorder. §(8 n – 3 n 4 + 10 n 2) – (3 n 2 + 11 n 4 – 7) § 8 n – 3 n 4 + 10 n 2 + -3 n 2 + -11 n 4 + 7 §-3 n 4 + -11 n 4 + 10 n 2 + -3 n 2 + 8 n + 7 §Combine like terms §-14 n 4 + 7 n 2 + 8 n + 7

How do I multiply monomials? §When multiplying, multiply the constants, then the variables. §Remember to use the laws of exponents (when multiplying you add the exponents). §Ex: 2 x 2 * 5 x 3 = 10 x 5

How do I multiply a monomial and a polynomial? §To multiply a monomial and a polynomial, you simply distribute. §Ex: 2 3 2 2 x (3 x – 4 x + x – 5) 6 x 5 – 8 x 4 + 2 x 3 – 10 x 2

How do I multiply a binomial by a binomial? §FOIL – multiply first, outside, inside, then last (basically distribute) §Box it – draw a box, put the numbers in, multiply and combine like terms. §Examples: § 1. (x + 2) (x + 5) § 2. (3 x + 10) (2 x – 5)

How do I multiply a polynomial by a polynomial? § 1) Distribute 2) Line up your like terms 3) Add §Or… Box it Ex: 1. (x + 2) (x 2 + 5 x + 6) 2. (x 2 – 2 x + 1) (x 2 + 5 x + 6) 3. (2 x 2 – 3 x + 4) (x 4 + 2 x 3 – 4 x – 3)

Divide a Polynomial by a Monomial Ex: (6 xy 2 – 3 xy + 2 x 2 y) ÷xy Is the same as: 6 xy 2 – 3 xy + xy 2 x 2 y §The 1 st way may be what you are used to seeing. The 2 nd way is written like a rational function. Just like fractions, we can split it up. Then we can follow the rules of exponents. § 6 xy 2 - 3 xy + 2 x 2 y xy xy xy § 6 y -3 + 2 x

Practice – Adding, Subtracting, Multiplying Polynomials and Dividing by a Monomial SHOW YOUR WORK On Notebook Paper: §Classwork: Odd Problems §Homework: Even Problems

Exit Ticket § 1. § 2. § 3. § 4. § 5. When do you add the exponents? When do you subtract the exponents? When do you multiply the exponents? How do you get 1 when using exponents? If something is raised to the 1/3 power, what is the root?