Polynomial Power-Up: Warm-Up!

Instructions: Think about words you know that start with the prefix "poly-". What do these words mean?

1. Word: Polygon
   What does it mean?
   
   
   
   

2. Word: Polyglot
   What does it mean?
   
   
   
   

3. Word: Polyculture
   What does it mean?
   
   
   
   

Think about it: Based on these words, what do you think the prefix "poly-" means?

Polynomial Power-Up Script

Warm-Up: What's in a Word? (5 minutes)

Teacher: "Good morning, class! To kick off our lesson today, I want us to think about a common prefix: 'poly-'. You've probably heard it in many words before. Look at the Polynomial Power-Up Warm-Up in front of you (or displayed on the board). Take about 2 minutes to write down what you think these words mean: Polygon, Polyglot, and Polyculture."

(Allow students time to think and write.)

Teacher: "Alright, who wants to share what they came up with for 'Polygon'?"

(Listen to student responses, guiding them to 'many sides' or 'many angles'.)

Teacher: "Excellent! How about 'Polyglot'? Any ideas?"

(Guide them to 'someone who speaks many languages'.)

Teacher: "Fantastic! And 'Polyculture'? This might be a new one for some of you."

(Guide them to 'the practice of growing many different crops together'.)

Teacher: "So, based on these examples, what do you think the prefix 'poly-' means?"

(Listen for 'many' or 'multiple'.)

Teacher: "You got it! 'Poly-' means 'many'. And that's a perfect lead-in to our topic today: Polynomials! As you might guess, 'polynomial' means 'many terms'. Let's power up our algebra skills and learn all about them!"

Introduction to Polynomials (10 minutes)

(Display Polynomial Power-Up Slide Deck - Slide 2: What's a Polynomial?)

Teacher: "So, what exactly is a polynomial? Simply put, a polynomial is an expression made up of one or more algebraic terms. Each term consists of a constant, which we often call a coefficient, multiplied by one or more variables raised to non-negative integer powers. This 'non-negative integer powers' part is really important – no square roots of variables, no variables in the denominator, and no negative exponents for our variables in a polynomial."

(Point to the definitions on the slide.)

Teacher: "Let's break down these important words:

• A term is a single number, a single variable, or numbers and variables multiplied together. Terms are separated by plus or minus signs.
• A coefficient is the numerical part of a term. It's the number in front of the variable.
• A variable is a letter, like x or y, that represents an unknown value.
• An exponent is that small number written slightly above and to the right of a base number or variable. It tells us how many times to multiply the base by itself."

(Display Polynomial Power-Up Slide Deck - Slide 3: Anatomy of a Polynomial)

Teacher: "Let's look at an example to see all these pieces in action: We have the expression $$ 3x^2 + 2x - 5 $$."

(Point to each part as you explain.)

Teacher: "First, can you identify the terms in this expression? Remember, they're separated by plus or minus signs."

(Wait for responses: 3x², 2x, -5.)

Teacher: "Great! Now, what about the coefficients? These are the numbers multiplying the variables, or the constant term itself."

(Wait for responses: 3, 2, and -5.)

Teacher: "Exactly! The constant term, -5, is also considered a coefficient. What are the variables we see here?"

(Wait for response: x.)

Teacher: "Yes, just 'x'. And finally, what are the exponents? Remember, if a variable doesn't have an exponent written, it's secretly a 1."

(Wait for responses: 2 and 1 for the x in 2x.)

Teacher: "Perfect! The exponent for x² is 2, and for 2x, the x has an invisible exponent of 1. The -5 technically has x⁰, which is 1, but we usually just say its exponent is 0."

Classifying Polynomials (8 minutes)

(Display Polynomial Power-Up Slide Deck - Slide 4: Classifying by Number of Terms)

Teacher: "Just like we classify animals or books, we can classify polynomials! One way is by the number of terms they have. And those 'poly-' prefixes we talked about come in handy here."

• "If a polynomial has only one term, we call it a monomial. Think 'mono' meaning one, like a monocle or monologue. Examples are 5x, -7, or 2y³."

• "If it has two terms, it's a binomial. 'Bi' means two, like a bicycle or binoculars. Examples: x + 4, 3y² - 2y, or 6a³ + 7b."

• "And if it has three terms? You guessed it, a trinomial. 'Tri' means three, like a tricycle or a triangle. Examples: x² + 2x - 1, 4a³ - 2a² + a, or y³ + 5y - 8."

Teacher: "What about polynomials with four or more terms? We generally just call them 'polynomials with four terms' or 'polynomials with five terms' and so on. The special names stop at trinomials for now."

(Display Polynomial Power-Up Slide Deck - Slide 5: What's the Degree?)

Teacher: "Another important characteristic of a polynomial is its degree. The degree of a term is simply the exponent of its variable. For example, in 5x³, the degree is 3. In 7x, the degree is 1. And for a constant like -12, the degree is 0 because there's no variable, or you can think of it as -12x⁰."

Teacher: "The degree of the entire polynomial is the highest degree among all of its terms. So, if we look at our earlier example, x² + 2x - 1, the degrees of the terms are 2, 1, and 0. The highest among those is 2. So, the degree of the polynomial x² + 2x - 1 is 2."

(Display Polynomial Power-Up Slide Deck - Slide 6: Practice Time!)

Teacher: "Let's try a couple together to make sure we've got it. Look at the first expression: 7x⁴ - 3x + 10.

• How many terms does it have? What are they?"
  (Wait for responses: 3 terms: 7x⁴, -3x, 10.)

Teacher: "Good! What are the coefficients?"
(Wait for responses: 7, -3, 10.)

Teacher: "Excellent! And what is the degree of this polynomial?"
(Wait for response: 4, because it's the highest exponent.)

Teacher: "Spot on! Since it has three terms, what kind of polynomial is it?"
(Wait for response: Trinomial.)

Teacher: "Fantastic! Let's do one more: -8y⁵.

• How many terms does it have? What is it?"
  (Wait for response: 1 term: -8y⁵.)

Teacher: "Perfect! What's the coefficient?"
(Wait for response: -8.)

Teacher: "You got it. And what's the degree?"
(Wait for response: 5.)

Teacher: "Great! Since it has only one term, what kind of polynomial is it?"
(Wait for response: Monomial.)

Teacher: "Wonderful job everyone! You're really getting the hang of this!"

Guided Practice: Worksheet Work (5 minutes)

(Display Polynomial Power-Up Slide Deck - Slide 7: Your Turn! Polynomial Practice)

Teacher: "Now it's your turn to put your new knowledge into practice! I'm going to hand out the Polynomial Practice Worksheet. We'll work on the first couple of problems together, and then you'll have a few minutes to work independently."

(Distribute the worksheets.)

Teacher: "Look at the first problem on your worksheet. Let's identify the terms, coefficients, degree, and then classify it. Who can tell me the terms in the first expression?"

(Guide students through the first 1-2 problems on the worksheet.)

Teacher: "Alright, you're doing great! For the remaining time, please work independently on the worksheet. I'll be walking around to answer any questions and help out. Remember to carefully identify all the parts we discussed today: terms, coefficients, variables, exponents, degree, and the classification by the number of terms. We will review these next class."

(Circulate and provide support. Collect worksheets as students finish or at the end of class.)

Polynomial Power-Up: Practice Worksheet

Instructions: For each expression below, identify the following:

• Terms
• Coefficients
• Degree (of the entire polynomial)
• Classification (Monomial, Binomial, or Trinomial)

1. $$ 5x^3 + 2x - 7 $$
   • Terms:
     
     
     
   • Coefficients:
     
     
     
   • Degree:
     
     
     
   • Classification:
     
     
     

2. $$ -9y^2 $$
   • Terms:
     
     
     
   • Coefficients:
     
     
     
   • Degree:
     
     
     
   • Classification:
     
     
     

3. $$ 4a + 12 $$
   • Terms:
     
     
     
   • Coefficients:
     
     
     
   • Degree:
     
     
     
   • Classification:
     
     
     

4. $$ 2m^4 - m^3 + 6m $$
   • Terms:
     
     
     
   • Coefficients:
     
     
     
   • Degree:
     
     
     
   • Classification:
     
     
     

5. $$ 15 $$
   • Terms:
     
     
     
   • Coefficients:
     
     
     
   • Degree:
     
     
     
   • Classification:
     
     
     

6. $$ 10p^2 - 3p^5 $$
   • Terms:
     
     
     
   • Coefficients:
     
     
     
   • Degree:
     
     
     
   • Classification:
     
     
     

Challenge Question!

Explain in your own words what makes an expression a polynomial and give an example of something that is not a polynomial.

Polynomial Power-Up: Practice Answer Key

1. $$ 5x^3 + 2x - 7 $$
   • Terms: 5x³, 2x, -7
   • Coefficients: 5, 2, -7
   • Degree: 3 (from 5x³)
   • Classification: Trinomial

2. $$ -9y^2 $$
   • Terms: -9y²
   • Coefficients: -9
   • Degree: 2 (from -9y²)
   • Classification: Monomial

3. $$ 4a + 12 $$
   • Terms: 4a, 12
   • Coefficients: 4, 12
   • Degree: 1 (from 4a)
   • Classification: Binomial

4. $$ 2m^4 - m^3 + 6m $$
   • Terms: 2m⁴, -m³, 6m
   • Coefficients: 2, -1, 6
   • Degree: 4 (from 2m⁴)
   • Classification: Trinomial

5. $$ 15 $$
   • Terms: 15
   • Coefficients: 15
   • Degree: 0 (constant term)
   • Classification: Monomial

6. $$ 10p^2 - 3p^5 $$
   • Terms: 10p², -3p⁵
   • Coefficients: 10, -3
   • Degree: 5 (from -3p⁵)
   • Classification: Binomial

Challenge Question!

Explain in your own words what makes an expression a polynomial and give an example of something that is not a polynomial.

Example Answer: An expression is a polynomial if it has one or more terms, where each term only has variables raised to whole number (non-negative integer) exponents. You won't see variables under a square root, in the denominator of a fraction, or with negative exponents in a polynomial.

Example of something NOT a polynomial:

• $$ \frac{1}{x} + 2 $$ (because x is in the denominator, which is like x⁻¹)
• $$ \sqrt{x} - 5 $$ (because x is under a square root, which is like x^(1/2))
• $$ 3x^{-2} + 4 $$ (because of the negative exponent)