Operations with Algebraic Expressions- Practice Problems and Solutions
What You Need to Know About Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and mathematical operations. If you can't work with them fluently, you're going to struggle through the rest of algebra. There's no way around it.
This guide cuts through the fluff. You'll get practice problems with solutions, explanations that actually make sense, and the common mistakes that cost people points.
The Four Operations You Must Master
1. Addition and Subtraction
You can only combine like terms. Like terms have the same variable raised to the same power.
Example:
3x + 5x = 8x ✓
3x + 5y = 3x + 5y ✗ (can't combine these)
7x² + 3x² - 2x = 10x² - 2x (combine x² terms, leave x alone)
2. Multiplication
Use the distributive property when multiplying a term by an expression in parentheses.
a(b + c) = ab + ac
Here's a multiplication example:
2x(3x + 4) = 2x·3x + 2x·4 = 6x² + 8x
When multiplying two binomials, use FOIL (First, Outer, Inner, Last):
(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
3. Division
Divide each term by the common factor. Simplify completely.
Example:
(12x² + 8x) ÷ 4x = 3x + 2
Check: 4x · 3x = 12x² ✓ and 4x · 2 = 8x ✓
Order of Operations Reminder
Always follow PEMDAS:
- Parentheses first
- Exponents (powers and roots)
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
Mess this up and you'll get wrong answers every time.
Comparing Operations: Quick Reference
| Operation | Key Rule | Example |
|---|---|---|
| Addition | Combine like terms only | 4x + 2x = 6x |
| Subtraction | Distribute negative sign first | 5x - 2x = 3x |
| Multiplication | Distribute to every term | 3(2x + 5) = 6x + 15 |
| Division | Divide each term by divisor | (8x²) ÷ 2x = 4x |
Practice Problems
Try these before checking the solutions. No peeking.
Problem 1: Simplify 4x + 7 - 2x + 3
Problem 2: Expand 3(2x - 5)
Problem 3: Multiply (x + 4)(x - 2)
Problem 4: Simplify (15x² + 10x) ÷ 5x
Problem 5: Expand and simplify 2(x + 3) + 4(x - 1)
Solutions
Solution 1
Group like terms:
4x - 2x + 7 + 3 = 2x + 10
Solution 2
Distribute the 3:
3 · 2x - 3 · 5 = 6x - 15
Solution 3
FOIL method:
(x + 4)(x - 2) = x² - 2x + 4x - 8 = x² + 2x - 8
Solution 4
Divide each term by 5x:
15x² ÷ 5x = 3x
10x ÷ 5x = 2
Answer: 3x + 2
Solution 5
Expand both, then combine:
2x + 6 + 4x - 4 = 6x + 2
Getting Started: Your Action Plan
- Identify like terms before adding or subtracting. Circle or highlight them.
- When subtracting, rewrite the expression with parentheses. Change every sign inside.
- When multiplying, work one term at a time. Don't try to do everything mentally.
- Check your work by substituting a simple number for the variable.
Where People Screw Up
- Forgetting to distribute the negative sign: 5x - (2x + 3) becomes 5x - 2x - 3, not 5x - 2x + 3
- Combining unlike terms: x + x² cannot be simplified. These are different terms.
- Skipping the distribution step in multiplication: always show your work.
- Misapplying FOIL: it only works for multiplying two binomials, not any two expressions.
One More Thing
Practice daily. Math isn't a spectator sport. You can read every solution in the world and still fail the test because you haven't trained your hands to do the work.
Do 10 problems a day. Check your answers. Fix your mistakes. That's it.