Operations with Fractions & Reducing Rational Expressions

Review rational expressions foundations. Math 20-1 Alberta Grade 11.

Lesson 6.1 of Rational Expressions and Equations in Math 20-1: Alberta curriculum lessons.

What you'll learn in this lesson

• Review: Operations with Fractions. Sharpen your fraction skills: adding, subtracting, multiplying, and dividing, using LCD, Keep-Change-Flip, and BEDMAS before applying them to rational expressions.
• Reducing Rational Expressions. Learn what a rational expression is, how to find non-permissible values, and how to reduce by cancelling common factors, not terms, after fully factoring numerator and denominator.

Inside the lesson: a free preview

The Three Operations with Fractions

1. Adding and Subtracting.

2. Multiplying.

3. Dividing, Keep, Change, Flip.

BEDMAS Reminder. When operations are mixed, follow BEDMAS: Brackets, Exponents, Division, Multiplication, Addition, Subtraction. Division and Multiplication have equal priority, so work left to right when they appear together.

What is a Rational Expression?

Definition: Rational Expression. A rational expression is an algebraic expression that can be written as the quotient of two polynomials, essentially a fraction with variables. Examples: (5x)/(15x), (x^2 + 2x + 1)/((3x+1)(x+1)).

The Golden Rule for Reducing. Reducing a rational expression works exactly like reducing a regular fraction: cancel common factors from the numerator and denominator.

Critical rule: You can only cancel factors (things being multiplied), never terms (things being added or subtracted). In (x+2)/(x-1) you cannot cancel the x's. They are part of larger terms, not standalone factors. You must factor first.

Non-Permissible Values (NPVs)

What are Non-Permissible Values?. A non-permissible value (NPV) is any value of the variable that would make the original denominator equal to zero. Since dividing by zero is undefined, those values are excluded.

How to Find NPVs: 3 Steps.

Important. Always identify NPVs from the original expression, before you reduce. A factor you cancel out can still create an NPV.

Worked example: a: Adding and Subtracting

Evaluate (9)/(5) + (2)/(7) - 3.

Part a) (9)/(5) + (2)/(7) - 3. Step 1: Find the LCD.

The denominators are 5, 7, and 1 (since 3 = (3)/(1)). Since 5 and 7 share no common factors, the LCD is 5 × 7 = 35.

Step 2: Rewrite each fraction with denominator 35.

(9)/(5) = (9 × 7)/(5 × 7) = (63)/(35)

(2)/(7) = (2 × 5)/(7 × 5) = (10)/(35)

3 = (3 × 35)/(1 × 35) = (105)/(35)

Step 3: Combine the numerators.

(63)/(35) + (10)/(35) - (105)/(35) = (63 + 10 - 105)/(35) = (-32)/(35)

Step 4: Check if it reduces.

32 = 2^5 (only factor of 2); 35 = 5 × 7 (no factor of 2). No common factor, already in lowest terms.

Answer: -(32)/(35)

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