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Polynomial Greatest Common Factor (Using the Distributive Property) Copyright©2000 Lynda Greene

Before we learn how to find a Greatest Common Factor there are two questions that usually come up. 1) What is a factor? 2) How do I find the factors of a number? Let’s use an example to answer both questions at the same time. NOTE: In this presentation we use the symbol ‘*’ to represent multiplication, we do this because when we have polynomials like 3x 3 + 4x 2 – x, we don’t want to confuse the multiplication symbol ‘x’ with the letter ‘x’.

Example: Say you want to find the factors of 8. The factors of 8 are all the numbers that will divide into 8 evenly. (In other words, they are not decimals like 2.38 or 4.1) So take the numbers from 1 to 8 and divide 8 by each of them. 1) 8 8) 8 2) 83) 84) 8 5) 86) 87) 8 What are factors and how do I find them? 8 42 decimal 1 If you get an answer with a decimal in it, the number you divided by is not a factor of 8, so cross out these answers. Now let’s look at the numbers that are left

1) 88) 82) 84) The numbers on top are the factors of 8, factors: 8, 4, 2 and 1 But, did you notice that the numbers on the top and the numbers you divided by (on the left) are the same? That’s because we are finding the factors two at a time, The number on the left and the number on top are both factors of 8. So to save time we don’t have to divide by every number from 1 to 8, we can go halfway and stop.

If we only have to find half of the factors, how do we know when we have gotten halfway and can stop? 1) Write the number with two little branches below it 8 2) Starting with ‘1 x 8’ Write all the pairs of factors that divide evenly into 8 1 * 8 2 * 4 4 * 2 8 * 1 3) This is where they start to repeat, STOP HERE! You don’t need to write these repeating numbers down If you write the factors of the number using the following system, you can see where your stopping point will be. All the factors of 8 are right here in this little box.

Practice: Find the factors of the following numbers * 12 2 * 6 3 * 4 1 * 32 2 * 16 4 * 8 1 * 81 3 * 27 9 * 9 1 * 48 2 * 24 3 * 16 4 * 12 6 * 8 7 * decimal Here’s where the numbers start to repeat 4 * 3, etc. so stop here. Factors of 12: 1, 2, 3, 4, 6, 12 5 * decimal 6 * decimal 7 * decimal 8 * 4(repeat) Make sure you check all the numbers up to the number on the bottom right, this is where they start to repeat. Factors of 32: 1, 2, 4, 8, 16, 32 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 81: 1, 3, 9, 27, 81 You can stop here since there are no more numbers between these two factors on the bottom We can stop checking numbers as soon as we reach this number Stop, since the next number is 8 Read the factors in this order Down the left side Up the right side

Now that we know what the factors of a number are and how to find them, we are ready to learn how to find the Greatest Common Factor.

The Greatest Common Factor (GCF)

We want: Greatest: The biggest number Common: all the terms have in common Factor: from a list of their factors In other words: 1) Find all the factors 2) Pick the biggest factor they all have in common (this is the GCF) * 12 2 * 6 3 * 4 1 * 6 2 * 3 1 * 9 3 * 3 Greatest Common Factor (GCF) The name Greatest Common Factor tells us what to do to the terms and also what we are looking for in the numbers GCF = 3

Find the GCF for these numbers: * 12 2 * 6 3 * 4 1 * 30 2 * 15 3 * 10 5 * 6 1 * 24 2 * 12 3 * 8 4 * 6 1 * 18 2 * 9 3 * 6 Notice that these numbers also have 1, 2 and 3 in common, but the Greatest Common Factor is the BIGGEST number they have in common, which is a ‘6’. GCF = 6

GCF for terms containing variables The little number on top of each x (the exponent), tells us how many x’s there are in each term. x 2 +x 3 +x4x4 x · x + x · x · x + x · x · x · x We want to take out (factor) as many of these x’s as we can. But, we have to take the same number of x’s out of each term.

x2x2 x3x3 x4x4 x · x + x · x · x + x · x · x · x The most we can factor out of all three terms is two x ’s (or GCF = x 2 ) 1. Write the GCF x2x2 2. Open Parentheses ( 3. Write the leftovers inside 1 + x + x 2 ) Note: The ‘1’ is explained on the next slide ++

x 2 (1 + x + x 2 ) When we factor out ‘everything’ in a term, there is still a ‘1’ in that spot. Notice that there is a ‘1’ in the first spot. re-multiplying x 2 is called: distributing You must be able to get the original expression back when you re-multiply the x 2. The reason? x 2 can be written as 1*x 2, so when you factor out the x 2 piece of it, the 1 is still there as a place-holder.

15x x 2 – 5x * 15 3 * 5 1 * 30 2 * 15 3 * 10 5 * 6 1 * 5 Most of the time, there is a mixture of numbers and variables (letters). So to find the GCF we have to find it in two parts. 1)GCF(of the numbers) 2)GCF(of the variables) GCF(numbers) = 5 GCF(numbers): 1) Find all the factors 2) Pick the biggest factor they all have in common

15x x 2 – 5x GCF = 5x Each term has at least ONE x in common So the GCF (variables) = ‘x’ II. GCF for the variables: pull out as many x’s as possible, but it has to be the same number of x’s from each term GCF(numbers) = 5GCF(variables) = x Put the two pieces together

So, now that we know the GCF of this polynomial is 5x, let’s factor it out. 1. Write the GCF 5x 2. Open Parentheses ( 3. Write the leftovers inside 3x 2 + 6x – 1) Note: We may be able to factor this polynomial further, but that is part of a different lesson (see factoring: the Greenebox Method) 15x x 2 – 5x

More Examples

2x 3 + 6x 2 – 4x 1 * 6 2 * 3 1 * 4 2 * 2 1 * 2 Another Example : 3. How many x’s do the 3 terms have in common? one ‘x’ 4. Put the two answers together. GCF = 2x 1. Write the GCF 2x 2. Open Parentheses ( 3. Write the leftovers inside x 2 + 3x – 2) Now factor it: 1. Factor all the numbers 2. What’s the largest number all 3 terms have in common? 2

Example: -3x 2 + 4x - 5 This negative must be removed (factored-out) (3x 2 - 4x + 5) This changes all the signs! the constant Important note: Technically, the polynomial should always be written in descending order. This means the terms go in order from the highest power to the lowest, the constant (the number with no “x”) is always written last. example: x 4 + x 3 + x 2 – x + 3. Also, the first term must be positive. If it is not, then factor out a ‘-1’.

3x 4 + 6x 3 – 4x 2 – 7x 1) Factor the numbers 1 * 6 2 * 3 1 * 4 2 * 2 1 * 3 1 * 7 2) What is the largest factor all 4 terms have in common? 3) Take out as many x’s as possible, how many x’s do all 4 terms have in common? If we take out a ‘1’, the other numbers do not change. So don’t bother, just say there is no numeric (number) GCF. Each term has at least one ‘x’, so the GCF = x 3x 4 + 6x 3 – 4x 2 – 7x So the overall GCF is ‘x’, now factor it! Recall: these little numbers (exponents, in red) tell you how many x’s are in each term. If there is no exponent, that’s an invisible number one 1 6x 3 – 4x 2 + 3x 4 – 7x These are not in the right order! Rearrange them in descending order

3x 4 + 6x 3 – 4x 2 – 7x 1. Write the GCF x 2. Open Parentheses ( 3. Write the leftovers inside 3x 3 + 6x 2 – 4x – 7 ) GCF = x

Practice Problems: (Hit enter to see the answers) 1)7x 2 – 7x ) 2y 2 – 6y 4 – 8y y 2)5x x 4 6) 2x 2 – 6x – 4 3) 18x 3 + 6xy – 9x 2 7) -2x 2 + 3x – 4 4) 12x x 4 – 24x 7 8) xyz + x 2 yz 3 – x 3 y 2 z Answers: 1) 7 (x 2 - x + 2) 5x 4 (x + 5) 3) 3x (6x 2 + 2y – 3x) 4) 4x 3 (3 + 4x – 6x 4 ) 5) 2y (y – 3y 3 – 4y 2 + 7) or 2y(-3y 3 - 4y 2 + y + 7) 6) 2 (x 2 – 3x – 2) 7) ( 2x 2 - 3x + 4) 8) xyz(1 + xz 2 – x 2 y)

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