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FACTORING ALGEBRAIC EXPRESSIONS Math 7 RECALL EXPANDING EXPRESSIONS Before, we used the Distributive Property to expand an algebraic expression. Recall that the two parts of the expression were called “factors”. For example, 3(2x + 9) Since the two things are being multiplied together (using the Distributive Property), the “3” and the “2x + 9” are said to be factors NEW TODAY Today, we will basically be doing the reverse of Expanding. The reverse of expanding is Factoring. Think about the result that you received on the previous slide when you expanded the expression 3(2x + 9)….6x + 27 What if I gave you 6x + 27 and asked you to get back to the “factored version?” In other words, get back to what it was prior to expanding it. FACTORING EXPRESSIONS Think about 6x + 27. Look at the two terms 6x and 27. What do you notice about both of them? Yes…they are both divisible by 3. Thus, we say that they share a common factor of 3. Since this is the case, you can “factor out” the 3 like this: 3( ) Since you factored out the three from both of the terms, what is left in each term? To find this, simply divide both terms by 3. FACTORING CONTINUED 6x + 27 and 3(2x + 9) are called equivalent expressions because they are equal to each other, but they look different. It is possible to have many different equivalent expressions to the same expression EQUIVALENT EXPRESSIONS For example: 12x + 24 has quite a few expressions that are equivalent to it. Can you give me some? Also, explain why they are equivalent 12(x + 2) is said to be the “simplest form” of 12x + 24 because the “greatest common factor” or “GCF” was factored out (12 is the largest factor that is in both 12 and 24) EXAMPLES Create an equivalent expression to the following by factoring out the “common term” 12x + 30 14 + 21x 72y – 36 6p – 45 MORE EXAMPLES 54y + 15x – 9 8m – 24 21p + 7 MAKE CERTAIN YOU ARE CORRECT How can we be sure that we have correctly created an equivalent expression? Use the Distributive Property to see if you get back to the original expression! This is a great way to “check” that you are correct. Go back and see if this works FACTORING WITH FRACTIONS Obviously fractions and mixed numbers will make things much more difficult. It will be more difficult because it is tough to recognize what they will actually have in “common” Take a look: 3 4 3 4 x2 x2 1 3 1 3 What do they have in common? FACTORING WITH FRACTIONS AND MIXED NUMBERS The first thing we should do is convert all mixed numbers to improper fractions. It is easier to deal with JUST FRACTIONS! 3 4 x2 1 would change to 3 x 7 3 4 3 Looking at the two fractions, what do they share? Would a common denominator help? This would make it 9 x 28 12 12 FACTORING WITH FRACTIONS AND MIXED NUMBERS Taking a look at 9 12 (___ ___) 28 you realize that they share…. x 12 12 9 How do we come up with the inside? 28 12 9 12 Divide, remember? So you must do 9 And you must do 9 12 12 28 9 12 DROP SWITCH FLIP x 12 x FACTORING WITH FRACTIONS AND MIXED NUMBERS Thus, the factored version would be: 3 1 (x 3 ) 4 9 In actuality, there are in infinite amount of equivalent expressions that we could come up with. We are essentially free to take out whatever we want, because we can always divide a fraction away from another fraction….lets take a look at this concept on the next slide. FACTORING WITH FRACTIONS AND MIXED NUMBERS 3 4 x 7 3 FACTORING WITH FRACTIONS AND MIXED NUMBERS On a state test, you are more likely to see these as multiple choice questions. If this is the case, the best way to approach them is to just expand all of the choices out to see which one will get you the original in return Example: Take a look at NYS Testing Program Sample Question…coming right up EXAMPLE FROM NYS TEST http://www.p12.nysed.gov/assessment/common-core- sample-questions/math-grade-7.pdf Another Example: 4 5 x2 1 4 is equivalent to which expression below? A) 4(5x + 2) B) 1 (16 x 8 ) 20 C) 2 5 (2 x 5 ) 5 8