Math help from the Learning Centre

Learning Objectives

The goal of this lesson is to 

• Identify like terms, monomials, and polynomials - noticing variations in their structure.
• Perform addition, subtraction, and multiplication of algebraic expressions - explore diverse forms and cases.
• Peform division of algebraic expressions - developing strategies through challenge.
• Perform long division of polynomials - applying reasoning and reflection through the process.

Explore Variation: Which Terms Are Like Terms?

Activity: Examine the following expressions. Decide which terms are like terms and explain your reasoning.

1. \(5x^2, -3x^2, 4xy, 7yx, 2x\)
2. \(6a^3b, 4ab^3, -2a^3b, 8a^2b\)

Why are \(4xy\) and \(7yx\) like terms, but \(6a^3b\) and \(4ab^3\) are not? What must be true about variables and exponents for terms to be like?

Take 5 minutes to discuss with a partner or write your thoughts before looking at the explanation (You can download solutions at the bottom of the page).

Variation in Operations

Example 1: Adding Like Terms

\[(3x^2+5x)+(7x^2-2x+4)\]

Example 2: Adding Unlikes Terms

\[(3x^2+5x)+(2xy-4)\]

• What is different about the results?
• Why can't we combine unlike terms?

Multiply Expressions with Different Structures

\[(x+2)(x^2-x+3)\]

Hints:

• Use distributive property carefully.
• Organize you work step-by-step.

Try first on your own, then compare with a partner.

Simple vs More Complex Division

Task 1: Divide the monomials:

\[\frac{12x^5}{3x^2}\]

Task 2: Divid the polynomials:

\[\frac{x^2-9}{x-3}\]

Notice how the first task is straightforward exponent simplification, but the second requires factoring. How does understanding the structure help in each case?

Variation in Dividend and Divisor Complexity

Guided Example: Divide

\(2x^3+3x^2-x+5\) by \(x+2\)

Step through the process slowly, reflecting on:

• How each term is processed.
• Why subtraction is necessary.
• What happens when you bring down terms.

Try dividing on you own:

\[(x^3-6x^2+11x-6) \div (x-2)\]

• Write each step.
• Identify where you get stuck.
• Reflect on strategies to overcome difficulties.

• Which parts of working with algebraic expressions felt most challenging?
• How did noticing variations help you decide what operations to perform?
• What strategies did you develop to persevere through difficult problems?

Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.