Lesson 8.5

Adding & Subtracting Polynomials

A polynomial is just a team of terms. To add teams, you simply combine the matching jersey numbers (like terms).

Introduction

We've mastered individual pieces (monomials). Now we put them together into Polynomials. Adding them is easy: it's just "Combining Like Terms" with a fancy name.

Past Knowledge

Lesson 2.7 (Like Terms). You already know that .

Today's Goal

Add complex polynomials vertically and horizontally.

Future Success

Necessary for Long Division of Polynomials in Pre-Calculus.

Key Concepts

Standard Form

Always write polynomials in Descending Order of power.

Correct: . Incorrect: .

Subtraction is Addition

The "Distribute the Negative" Trap

When subtracting a polynomial in parentheses, you MUST flip every sign inside.

Worked Examples

Example 1: The Horizontal Method

Basic

Add:

1

Group Like Terms

Squares with Squares. X's with X's.

2

Combine

and .

Answer

Example 2: Subtraction

Intermediate

Simplify:

1

Distribute Negative

Flip signs of the SECOND group only.

2

Combine

Squares:

Linears:

Constants:

Answer

Example 3: Missing Terms

Advanced

Add:

1

Notice Mismatches

The first has cubes, no squares. The second has squares, no cubes. Nothing combines!

2

Order Them

Just write them in descending order.

Answer

Common Pitfalls

Adding the Exponents

When Adding, exponents STAY THE SAME. is , NOT . Only multiply exponents when... multiplying.

Incomplete Subtraction

Students often flip the first sign but forget the rest. Distribute the negative to everyone in the second parenthesis.

Real-Life Applications

Revenue vs. Cost:

  • Profit = Revenue - Cost.
  • If Revenue is and Cost is ...
  • We must subtract the polynomials to find the Profit function.

Practice Quiz

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