ࡱ> 8: 7tg ,bjbjVV 4\r<r<$thLZ$v****#######$&(b#|#**$(**##:#,#*uE# #*$0Z$O#R))#)#$##Z$) : Understanding Factoring To factor a quantity means that it is broken up into numbers or expressions that could be multiplied to produce the original problem. Those numbers are called factors of the original problem. Example: 6 and 4 are called factors of 24 because 6 * 4 = 24. However, usually these problems are broken up (factored) until they will not break up (factor) any further. They are then said to be factored completely so 24 completely factors into 2*2*2*3. You might recall that numbers like 2, 3, 5, 7, 11, etc. are called prime numbers which means they cant be factored any further. You probably have worked with factor trees before. Look at the example below.  EMBED Equation.DSMT4  18   2 9 so 18 = 2 * 3 * 3 (since 2 and 3 are prime numbers, 18 wont factor any further and is said to be completely factored) 3 Test your understanding: factor each of the following. 1) 108 2) 51 3) 750 In algebra II, we typically work with expressions that need to be factored. Sometimes they are just single terms (no addition or subtraction is shown) and so they are called Monomials. An example is  EMBED Equation.DSMT4 . Test your understanding: factor  EMBED Equation.DSMT4  Sometimes we work with expressions that involve 2 or more terms (which means expressions that are separated by either a subtraction sign or an addition sign that is NOT located inside of any parentheses). Examples are listed below. 1)  EMBED Equation.DSMT4 (since this problem has one term of x and another of 5y joined by a subtraction sign, it is called a Binomial 2)  EMBED Equation.DSMT4  (since this problem has one term of  EMBED Equation.DSMT4 , another of  EMBED Equation.DSMT4 , and yet a third one thats  EMBED Equation.DSMT4 , it has three terms and it is called a Trinomial). 3)  EMBED Equation.DSMT4 (since this problem has 4 terms, it is just called a polynomial with 4 terms). Note that all examples given above are called polynomials too, but for problems that have as many as 4 terms we just dont give them special names. Test your understanding: Complete the table below. Remember that terms have to be separated by either addition or subtraction signs that are NOT inside of parentheses. The Given ProblemHow many terms does it have?The name of this type of polynomial is what? EMBED Equation.DSMT4 55 EMBED Equation.DSMT4  EMBED Equation.DSMT4 (x 5)(3x + 87) A critical issue in learning to factor polynomials is an understanding of Greatest Common Factor (often abbreviated as GCF). A GCF is an expression (monomial) that consists of the product of all factors that each term in a polynomial has in common. For example:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  could each be separately factored into  EMBED Equation.DSMT4 =2*2*3*x*x*y*y*y and  EMBED Equation.DSMT4 = 2*2*2*x*y*y. What is the most that these problems have in common? They both have 2s, at least one x, and at least two ys. That makes the GCF = 2*2*x*y*y =  EMBED Equation.DSMT4 . So  EMBED Equation.DSMT4  is called the GCF. Test your understanding: Find the GCF of  EMBED Equation.DSMT4  One thing that you might find difficult about learning to factor polynomials is that there is no magic formula and only minimal help available from an appropriate calculator. However, there is ONE RULE which ALWAYS takes precedence over all other factoring strategies. It is: ALWAYS FACTOR OUT THE GREATEST COMMON FACTOR WHENEVER POSSIBLE. What is meant by factor out? Lets take that last problem and join them together with an addition sign to make a binomial. It would be  EMBED Equation.DSMT4 To factor out the GCF from this expression means to put outside a set of parentheses the GCF of the two terms and to put inside the parentheses whatever is left over when each of the factors in the GCF is removed from the original terms. That makes it look like this.  EMBED Equation.DSMT4 = 2*2*3*3*x*x*x*y + 3*3*5*x*x*y*y = 3*3*x*x*y (2*2*x + 5*y) =  EMBED Equation.DSMT4  This last step shows the answer with the GCF factored out. Test your understanding: Factor out the GCF from  EMBED Equation.DSMT4  As you learn other problem patterns later, keep in mind that there is one thing you should ALWAYS do and that is to factor out the GCF first! Problems that contain four terms are usually factored by a method called factoring by grouping. In this type of problem we begin by putting parentheses around (grouping) the first two terms and around the second two terms leaving either an addition or a subtraction sign between the parentheses. Example:  EMBED Equation.DSMT4  has 4 terms so well insert the parentheses to make it look like this:  EMBED Equation.DSMT4  Next we factor out the GCF from each of the parentheses separately. Since the GCF of  EMBED Equation.DSMT4 is 2x and since the GCF of 21x and 35 is 7 we can rewrite the problem as  EMBED Equation.DSMT4 . Notice how the expressions inside the parentheses match? If they didnt match weve either made a mistake or else this method of factoring by grouping is not useful on this problem. The fact that they do match leads us to the last step on this problem. Inside one set of parentheses put the terms that match and inside the other parentheses put the other terms which are outside the parentheses. This gives our final answer of  EMBED Equation.DSMT4 . Important! If it happens that the subtraction sign falls between the 2 parentheses when we first add them at the beginning of the problem, then we must change the sign inside the second parentheses! Look at the example below to notice this and how the other steps are the same as the example above. Example:  EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Problems that contain three terms are factored by either trial and error or using a variation of factoring by grouping. Your instructor will give you information about this last type of problem. No matter what type of problem you are trying to factor, always remember to factor out the GCF first if its possible, then, and only then try to use the patterns shown above. Also remember this sheet is just a summary intended to get you started in recognizing the various patterns of factoring. After you have a few days to practice all the problem types you will be on your own so no notes will be allowed on quizzes or tests. THE ONLY WAY TO LEARN HOW TO FACTOR IS TO PRACTICE! Special Patterns for Factoring: A key to success at factoring polynomials is recognition of the pattern that a problem fits. This sheet is a summary of a few basic patterns (there are others). After practice these patterns should become easy to recognize and will need to be memorized. Problems that contain two terms usually fit one of the 4 patterns below. The difference of two squares consists of 2 terms that are separated by a subtraction sign. Additionally, each is the square of some expression. Example:  EMBED Equation.DSMT4 has two terms separated by a subtraction sign and each term is something squared. This pattern always factors into two binomials in parentheses with different signs in the middle. That is,  EMBED Equation.DSMT4 . Note: Sometimes youll have to write the terms as squares first! Example:  EMBED Equation.DSMT4  Then that can be factored into (2a 5b) (2a + 5b). The sum of two squares consists of 2 terms that are separated by an addition sign. Additionally, each is the square of some expression. Example:  EMBED Equation.DSMT4  has two terms separated by an addition sign and each term is something squared. This pattern does not factor so is said to be prime. The difference of two cubes consists of 2 terms that are separated by a subtraction sign. Additionally, each is the cube of some expression. Example:  EMBED Equation.DSMT4  has two terms separated by a subtraction sign and each term is something cubed. This pattern always factors into the pattern shown.  EMBED Equation.DSMT4  Note: Sometimes youll have to write the terms as cubes first! Example:  EMBED Equation.DSMT4  can be first written as  EMBED Equation.DSMT4 . Then that can be factored into  EMBED Equation.DSMT4  which simplifies to make  EMBED Equation.DSMT4  The sum of two cubes consists of 2 terms that are separated by an addition sign. Additionally, each is the cube of some expression. Example:  EMBED Equation.DSMT4  has two terms separated by an addition sign and each term is something cubed. This pattern always factors into the pattern shown.  EMBED Equation.DSMT4  Note: Sometimes youll have to write the terms as cubes first! Example:  EMBED Equation.DSMT4  can be first written as EMBED Equation.DSMT4  Then that can be factored into  EMBED Equation.DSMT4  which simplifies to make EMBED Equation.DSMT4 .  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