Add, Subtract, and Multiply Polynomials

Lesson 1 of 7: Arithmetic with Polynomials

In this lesson:

• Add and subtract polynomials by combining like terms
• Multiply polynomials using the distributive property
• Explain why polynomials are closed under these three operations

Learning Objectives for Today's Lesson

1. Add and subtract polynomials by combining like terms
2. Multiply polynomials using the distributive property
3. Explain why polynomials are closed under , , but not
4. Identify the degree and leading coefficient of a product
5. Verify operations by evaluating both sides at a specific value

Recall: Polynomials and Their Vocabulary

• A polynomial: terms with non-negative integer exponents
• Monomial (1 term): — Binomial (2 terms):
• Degree: highest exponent — has degree

Can you recognize a polynomial when you see one?

Three Operations Stay Inside, One Escapes

Same for polynomials?

Predict What Polynomials Give You

What type does each operation give?

Closure: Polynomials Stay in Their System

Polynomials are closed under addition, subtraction, and multiplication.

• Adding two polynomials → always a polynomial ✓
• Subtracting two polynomials → always a polynomial ✓
• Multiplying two polynomials → always a polynomial ✓
• Division can escape: ✗

Like Terms: The Foundation of Addition

Like terms have identical variable parts — same variable, same exponent.

• and are like terms (both )
• and are not like terms — different exponents
• and are not like terms — different variables

Adding polynomials means combining like terms only.

Adding Polynomials Step by Step

Add:

Step 1: Group like terms

Step 2: Combine

Degree: 3 ✓ — leading term is

Subtract by Distributing Negative One

— every sign in the second polynomial flips:

Find the Sign Error Here

A student subtracted and got:

Which term is wrong, and why?

Identify the error before advancing.

Subtract and Check Your Answer

Subtract:

Result:

Verify at :

• Original:
• Result: ✓

Quick Check: Addition and Subtraction

Simplify:

Distribute the negative, then combine like terms.

1. What sign does the term take after distribution?
2. What is the final simplified expression?

Addition Combines — Multiplication Expands

• Adding: — sort and combine like terms
• Multiplying: — every term × every term

Multiplication uses the distributive property, applied to polynomials.

It never fails, no matter how many terms you have.

Distributive Property Powers All Multiplication

— distribute across :

FOIL: Binomial × Binomial Only

First , Outer , Inner , Last →

When FOIL Misses Three Products

— FOIL gives only First and Last; it misses , ,

Distribute each row instead:

Multiplying with the Table Method

— organize using a table:

Sum all cells:

Degree of a Product: Predict It First

• Degree of product = sum of degrees
• Leading term = product of leading terms

Always Expand Squared Binomials Fully

— the middle term is missing!

Check at : but ✗

Predict Degree and Leading Term

Without computing, predict for :

1. What is the degree of the product?
2. What is the leading term?

Apply the degree rule — predict before computing.

What Changes When a Sign Flips?

We computed:

What if the second factor becomes instead?

• Which products change?
• Does the degree change?
• Does the sign of the constant term change?

Reason about this before computing the full product.

Mixing All Three Operations Together

• Add/Subtract: combine like terms
• Multiply: distribute every term × every term
• Mixed: multiply first, then combine

Order of operations still applies — expand products before collecting terms.

Mixed Operations: Two Squared Binomials

Simplify:

• Subtract: and constant terms cancel →

Verify at : ✓

Mixed: Expand First, Then Subtract

Simplify:

Expand:

Subtract and collect:

Simplify This Completely on Your Own

No hints — expand, distribute, and combine.

State the degree and leading coefficient of your answer.

Is It Still a Polynomial?

Reflection: Is a polynomial? Why or why not?

A polynomial requires:

• Non-negative integer exponents
• Real number coefficients
• Finite number of terms

Apply the definition. Where does fail it?

Watch for These Three Common Errors

Sign flip: — every term flips, not just the first

FOIL scope: needs full distribution — FOIL misses products

Unlike terms: — can't combine different degrees

What You Can Now Do with Polynomials

✓ Closed under , , — always gives a polynomial

✓ Subtract: distribute to every term in the second polynomial

✓ Multiply: distribute every term; FOIL for binomials only

Division breaks closure — rational expressions come later

Next Lesson: Division and the Remainder

Division breaks closure — but the remainder carries information.

Dividing by : the remainder equals the polynomial's value at .

APR.B.2 — The Remainder Theorem