Lesson 1 – Operations with Polynomials 10. Polynomial Function ~ relates an input to I. Vocabulary an output, three main parts

1. Constant ~ is a number (on its own)

2. Term ~ is either a single number or a the input variable, or numbers & variables multiplied the relationship together. . . terms are separated by +/- the output

3. Polynomial~ comes from poly- (meaning 11. Factors ~ are numbers you can multiply “many”) and –nomial (in this case meaning together to get another number, in “term”) . . . so it means “many terms” they algebra factors are what you can can have constants, variables, and can multiply together to get an exponents but NEVER division by a variable.

4. Monomial ~ is a polynomial with 1 term II. Add/Subtract Polynomials

5. Binomial ~ is a polynomial with 2 terms *COMBINE LIKE TERMS!

6. Trinomial ~ is a polynomial with 3 terms 1. 3𝑥! + 2𝑥! − 𝑥 − 7 + 𝑥! − 10𝑥! + 8

7. Leading Coefficient ~ is the coefficient of Degree = the first term of a polynomial LC =

8. Degree ~ (of a polynomial) with only one 2. 8𝑥! − 3𝑥! − 2𝑥 + 9 − 2𝑥! + 6𝑥! − 𝑥 + 1 variable is the largest exponent of that Degree = variable LC =

9. Standard Form (a.k.a. descending order) 3. 2𝑥! + 3𝑥 − 2𝑥! + 3𝑥! + 𝑥 − 4 for writing down a polynomial is to put the Degree = terms with the highest degree first LC = i.e. 3𝑥! − 7+ 4𝑥! + 𝑥! the highest degree is 6, next is 3, then 2, and last the constant 𝑥! + 4𝑥! + 3𝑥! − 7

Page Page

4. 3𝑥 + 1 + 𝑥! + 2𝑥! − 4 − 𝑥! − 4𝑥! + 7𝑥 You Try: (work these out on page 60) Degree = 1. 𝑥! + 2𝑥 + 3 𝑥! − 4𝑥 + 5 LC = 2. 𝑥 + 2 3𝑥 + 2 2𝑥 + 4

III. Multiplying Polynomials 3. 2 𝑥! + 5𝑥 − 1 𝑥! − 2𝑥 + 1 Choose Method ~ DISTRIBUTE or BOX IV. Evaluating Polynomial Functions: 1. 2𝑥 − 1 3𝑥 + 4

Function: a relation for which each value from the domain (input) is paired with exactly one value in the range (output). *must pass

2. −𝑥! + 2𝑥 + 4 𝑥 + 3 the vertical line test!

Domain: the input values (x-values) Range: the output values (y-values)

3. 𝑥! − 3 3𝑥! − 2𝑥 − 4 Evaluate a function: to replace the variables

with a number or expression

4. 𝑥 − 1 𝑥 + 4 𝑥 + 3 Evaluate each for the given values: 1. 𝑓 𝑥 = 𝑥 𝑓 3 𝑓 0 𝑓 −2

2. 𝑓 𝑥 = 3𝑥! − 4 𝑓 3 𝑓 0 𝑓 −2 5. 𝑥 − 2 𝑥 + 4 𝑥 + 2

3. 𝑓 𝑥 = −4𝑥! + 2𝑥 𝑓 3 𝑓 0 𝑓 −2

Page Page

4. 𝑔 𝑥 = 5𝑥! + 4𝑥! − 5 4. 𝑔 𝑥 = 5𝑥! + 4𝑥! − 5 3 𝑔 2 𝑔 2 + 5 −2𝑔 2 − 7 3 𝑔 2 𝑔 2 + 5 −2𝑔 2 − 7

5. 𝑓 𝑥 = 4𝑥 + 3 𝑓 2𝑦 5. 𝑓 𝑥 = 4𝑥 + 3 𝑓 2𝑦 6. 𝑓 𝑥 = 3𝑥 − 7 𝑓 𝑥 − 4 6. 𝑓 𝑥 = 3𝑥 − 7 𝑓 𝑥 − 4 7. 𝑓 𝑥 = 3𝑥! − 2𝑥 + 5 𝑓 𝑥 + 2 7. 𝑓 𝑥 = 3𝑥! − 2𝑥 + 5 𝑓 𝑥 + 2 8. 𝑓 𝑥 = −𝑥! + 6𝑥 − 2 𝑓 −4𝑥 8. 𝑓 𝑥 = −𝑥! + 6𝑥 − 2 𝑓 −4𝑥 9. 𝑓 𝑥 = 𝑥! − 9𝑥 + 8 𝑓 𝑥 − 1 9. 𝑓 𝑥 = 𝑥! − 9𝑥 + 8 𝑓 𝑥 − 1

INPUT

INPUT

INPUT

INPUT

INPUT

Page Page 61

INPUT

INPUT

INPUT

INPUT

INPUT