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Algebra I – Algebraic Fractions

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    Study Guide Algebra I – Algebraic Frac Ɵ ons 1. What Are Algebraic Frac Ɵ ons? Algebraic fractions are just like normal fractions — but they include variables (letters like x, a, or b). For example: Here, x is a variable in the denominator. So, an algebraic fraction is any fraction that has: • A variable in the numerator, or • A variable in the denominator, or • Variables in both. Why Do We Have Restrictions? You already know something very important: Division by 0 is impossible. Because of this, any value that makes the denominator equal to 0 is not allowed. That means: The denominator of a fraction can never be zero. So whenever we see an algebraic fraction, we must check what values would make the denominator zero — and exclude them. These excluded values are called restrictions .

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    Study Guide Let’s Look at Some Examples 1. Example: 5/x Since x is in the denominator: Why? Because if x = 0, the fraction becomes: And that is undefined. So the restriction is: x cannot equal 0 2. Example: Here the denominator is x − 3 . To find the restriction, ask: When does the denominator equal 0?

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    Study Guide So, x cannot equal 3 We write: 3. Example: Now the denominator is a − b . Set it equal to 0: [a - b = 0] [a = b] So, a cannot equal b We write: 4. Example: The denominator is: [a 2 b] This expression equals 0 if:

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    Study Guide • (a = 0), or • (b = 0) So neither value is allowed. a ≠ 0 and b ≠ 0 Both variables must be nonzero. 2. Opera Ɵ ons with Algebraic Frac Ɵ ons Introduction When working with algebraic fractions , the steps are very similar to working with regular fractions. The difference is that now we are working with variables (like x ) along with numbers. In this chapter, you’ll learn how to: • Reduce algebraic fractions • Multiply algebraic fractions • Avoid common mistakes As you go through the examples, pay close attention to the steps. Learning the correct method will save you time and help you avoid errors. Reducing Algebraic Fractions Reducing means simplifying a fraction to its lowest terms. Step - by - Step Method: 1. Factor the numerator (top part). 2. Factor the denominator (bottom part).

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    Study Guide 3. Cancel common factors (only factors — not individual terms). 4. Write the simplified result. Very Important: You can only cancel factors , not terms that are being added or subtracted. Example 1 1) Simplify: Step 1: Reduce the numbers 4 ÷ 8 = 1/2 Step 2: Reduce the variables Final Answer: 2) Simplify: Step 1: Factor both numerator and denominator

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    Study Guide Step 2: Cancel the common factor ((x - 1)) Final Answer: Notice we canceled the entire factor (x − 1) — not just part of it. 3) Simplify: Step 1: Factor both parts Numerator: Denominator: Step 2: Cancel the common factor ((x + 1)) Final Answer : Warning: A Common Mistake

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    Study Guide You cannot reduce across addition or subtraction. For example: You cannot cancel the x’s. Also: You cannot cancel the 6 with part of the numerator. Always factor first. If something is being added or subtracted, do not cancel unless it is part of a common factor. Multiplying Algebraic Fractions Multiplying algebraic fractions is actually easier than it looks! Follow these steps: 1. Factor all numerators. 2. Factor all denominators. 3. Cancel common factors. 4. Multiply what remains on top. 5. Multiply what remains on bottom. If you reduce correctly before multiplying, your final answer will already be simplified. Example 2: Multiplying Algebraic Fractions

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    Study Guide How to Multiply Algebraic Fractions Multiplying algebraic fractions is easier than it looks! Just follow these simple steps: Step - by - Step Method: 1. Factor everything first (numerators and denominators). 2. Cancel common factors . 3. Multiply what remains on top . 4. Multiply what remains on bottom . 5. Simplify your final answer if needed. 1) Multiply: Step 1: Multiply numerators Step 2: Multiply denominators Final Answer: Simple — no common factors to cancel!

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    Study Guide 2) Multiply: Step 1: Multiply across Step 2: Cancel common factors • (y) cancels with (y) • One (x) cancels from (x 2 ) Final Answer: Always cancel before multiplying completely — it keeps things easier! 3) Multiply: Step 1: Factor everything

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    Study Guide Step 2: Rewrite with factors Step 3: Cancel common factors Cancel: • ((y+2)) • one ((x+1)) Final Answer: 4) Multiply: Step 1: Factor expressions Step 2: Rewrite

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