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7.3 Reduce Rational Expression Before we start, let’s look at some definitions… A Rational (Rational Expression): A fraction that has a polynomial (something with a variable) in the Denominator. These are Rationals… 5 2x x 2 + 65 3x + 7 These are not… x 2 + 2x + 8 2 5x 2 + 8 17 What’s the difference between these? The denominator must have a variable to make a rational expression.

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7.3 Reduce Rational Expression Since a rational expression is really just a fraction, we use the rules of fraction to work with them… 25x 2 5x = Here’s how… Hint: ÷5 5x 2 x Fractions can be added, subtracted, multiplied and divided Fractions can be reduced: Numbers with Numbers and Variables with Variables. Great Job! Now for Division Rules for Exponents (Boss Rule) x 2 is boss and stays (x 2-1 = ?) = 5x

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7.3 Reduce Rational Expression Sometimes the problems get a little more complicated. Here is a rule you will have to use throughout MT7. Here’s what I mean… Never, ever, ever, ever cancel into a parenthesis! You must factor first! Parenthesis exist whenever you have terms (+ or – signs), whether you see them or not. Parenthesis are there whether you see them or not! 3x + 6 is really (3x + 6) and x 2 + 5x – 14 is really (x 2 + 5x – 14) The only way to reduce a rational is the factor whenever possible. What does factor mean again? To make ( )

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7.3 Reduce Rational Expression We have learned 3 types of factoring. They are… 1. GCF 2. Difference of 2 Squares 3. Diamond Method. 3x + 6 3(x + 2) 4x 2 – 9 (2x + 3)(2x – 3) x 2 – 9x + 20 (x - 4)(x – 5) You must master all three methods if you are going to be successful in MT7. This whole topic requires you to factor then cancel over and over again. Make sure you are good at it.

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7.3 Reduce Rational Expression Your problems will look like this… Example: 5x + 10 x 2 - 2x - 8 First GCF the Numerator And Diamond Method the Denominator 5(x + 2) (x – 4)(x + 2) Need help with factoring? Ask your teacher or look online under factoring. Now cancel the (x + 2)’s and you are done! 5 (x – 4) The Answer! Remember: You can never, ever cancel INTO parenthesis, but you can cancel the parenthesis if they are exactly the same.

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7.3 Reduce Rational Expression And another one… Example: 3x 2 + 12x 6x First GCF the Numerator And nothing happens to the Denominator 3x(x + 4) 6x Now cancel the 3 and 6 and the x’s. (x + 4) 2 The Answer! 1 2

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