Polynomials Sec 9.1.1

Learning Targets Vocabulary Operations between polynomials Introduction to graphs of polynomials

Definitions Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it means “many terms” Term: A number, a variable, or the product/quotient of numbers/variables.

Polynomial Example of Polynomial 𝑦= 5𝑥 3 + 0.5𝑦 2 − 19𝑥𝑦 3 −7 Terms

5 𝑥 3 A Term has 3 Components: Exponent: Can only be positive integers: 0,1,2, 3, Coefficient: can be any real number… including zero. Variable These components are very important!!!

NOT ALLOWED Negative exponents: 𝑥 −2 Variables in the denominator: 4 3−𝑥

Check In Which of the following is a polynomial: −6𝑦2 − 7 9 𝑥 −6𝑦2 − 7 9 𝑥 4𝑥− 1 3 𝑦+ 𝑥𝑦 −3 −41𝑥𝑦 3𝑥𝑦𝑧 + 3𝑥 𝑦2𝑧 − 0.1𝑥𝑧 − 200𝑦 + 0.5 5

Naming a Polynomial 7 5x + 2 4x2 + 3x - 4 6x3 - 18 1 2 3 monomial We can classify a polynomial based on how many terms it has: Polynomial 7 5x + 2 4x2 + 3x - 4 6x3 - 18 # Terms 1 2 3 # Terms Name monomial binomial trinomial

Naming Cont. Quadrinomial (4 term) and quintinomial (5 term) also exist, but those names are not often used. Polynomials Can Have Lots and Lots of Terms Polynomials can have as many terms as needed, but not an infinite number of terms. For more than 3 terms say: “a polynomial with n terms” or “an n-term polynomial” 11x8 + x5 + x4 - 3x3 + 5x2 - 3 “a polynomial with 6 terms” – or – “a 6-term polynomial”

The degree of a term is determined by the exponent of the variable. 3 4x -5x2 18x5 Degree of Term 1 2 5 New vocabulary! It is a simple concept. It is just the exponent of the variable. You can find the degree of multiply variable terms, but we don’t deal with them in algebra 1. Now these degrees are very important. They tell you which family of functions a term belongs to and will have a big impact on polynomials.

Naming a Polynomial 7 5x + 2 4x2 + 3x - 4 6x3 - 18 1 2 3 Constant We can also classify a polynomial based on its highest degree: Polynomial 7 5x + 2 4x2 + 3x - 4 6x3 - 18 Degree 1 2 3 # Degree Name Constant Linear Quadratic Cubic

Putting it All Together Polynomial -14x3 -1.2x2 -1 7x - 2 3x3+ 2x - 8 2x2 - 4x + 8 x4 + 3 Name cubic monomial quadratic monomial constant monomial linear binomial cubic trinomial quadratic trinomial 4th degree binomial

Standard Form of a Polynomial A polynomial written so that the degree of the terms decreases from left to right and no terms have the same degree. Very much common sense. Basically Standard form is simplified (like terms are combined), you just have to make sure to write the terms in the correct order. The highest degree comes first… the rest follow in order of decreasing degree.

Not Standard 6x + 3x2 - 2 15 - 3x - x+ 5x4 x + 10 + x 1 + x2 + x + x3 Examples

Operations Polynomials can be added, subtracted, multiplied and/or divided The following slides will cover addition, subtraction and multiplication We will learn about division later on in the unit

Adding and Subtracting Polynomials To add or subtract polynomials, simply combine like terms. (5x2 - 3x + 7) + (2x2 + 5x - 7) = 7x2 + 2x (3x3 + 6x - 8) + (4x2 + 2x - 5) = 3x3 + 4x2 + 8x - 13 So, nothing really new here. We’ve been doing this for a long while. I’m just going to add more terms than you are used to and terms of higher degree. (2x3 + 4x2 - 6) – (3x3 + 2x - 2) (2x3 + 4x2 - 6) + (-3x3 + -2x - -2) = -x3 + 4x2 - 2x - 4

Polynomial Multiplication To multiply polynomials we must distribute all of the terms Ex: 𝑥 3 + 4𝑥 2 +1 × − 3𝑥 2 −2𝑥 𝑥 3 + 4𝑥 2 +1 −3 𝑥 2 −3 𝑥 5 −12 𝑥 4 −3 𝑥 2 Combine Like Terms! −2𝑥 −2 𝑥 4 −8 𝑥 3 −2𝑥 −3𝑥 5 − 14𝑥 4 − 8𝑥 3 − 3𝑥 2 −2𝑥

Polynomial Multiplication Multiply the following polynomials:

Polynomial Multiplication (x + 5) x (2x + -1) -x + -5 2x2 + 10x + 2x2 + 9x + -5 (3w + -2) x (2w + -5) -15w + 10 + 6w2 + -4w 6w2 + -19w + 10

Polynomial Multiplication (2a2 + a + -1) x (2a2 + 1) 2a2 + a + -1 + 4a4 + 2a3 + -2a2 4a4 + 2a3 + a + -1

Investigating Graphs of Polynomials Pg. 437 In your notes go through problem 9-1 silently… Write down any conjectures, similarities or patterns you see After 5 minutes we will discuss in our teams

For Tonight Homework: Pg. 440: 9-8 9-11, 9-13, 9-14 and 9-18 Answers to these questions will be posted online tonight