By the end of this lesson, you will be able to:
Use all factoring tools to simplify algebraic fractions by identifying and canceling algebraic primes.
Perform multiplication and division of algebraic fractions by factoring and canceling common factors.
Apply the LCD procedure to add or subtract algebraic fractions by factoring and identifying least common denominators.
Below are two fractions. Without simplifying, what do you notice?
| Expression A | Expression B |
| \[\frac{x^2-9}{x^2-x-6}\] | \[\frac{x^2+5x+6}{x^2+2x-3}\] |
Prompt:
What similarities in structure do you notice?
Which expressions might be factored?
How does the numerator relate to the denominator?
Takeaway: Simplifying algebraic fractions always starts with factoring — and recognizing patterns is key.
Explore the following:
\[\frac{x^2-4}{x^2+2x}\]
\[\frac{3x^2+6x}{x^2+3x}\]
\[\frac{x^2+5x+6}{x^2+2x}\]
Prompt:
What can and can’t be canceled?
Why must you factor first?
📣 Tip: Cancel only full factors, not individual terms!
Multiply straight across, but always factor first
For division: Flip the second fraction (reciprocal), then multiply
Try the following:
\[\frac{x^2-25}{x^2+5x}\cdot\frac{x}{x+5}\]
\[\frac{x^2+3x}{x^2-16}\div\frac{x+4}{x}\]
🧩 Compare and Discuss:
Which factors canceled?
What factoring method was most useful?
🧠Prompt:
How does the structure of each expression inform your strategy?
Factor all denominators
Identify the LCD (least common multiple of all factors)
Rewrite each fraction with the LCD
Add or subtract
Simplify final expression if possible
Try the following:
\[\frac{1}{x+2}+\frac{2}{x^2-4}\]
\[\frac{3x}{x^2+4x+4}-\frac{2}{x+2}\]
Can the LCD sometimes already be a denominator?
How do you avoid common errors in subtraction?
Prompt:
Which factoring tools did you use the most?
Where do you feel most confident?
What differences between multiplication, division, and addition/subtraction stood out to you?