C ollege A lgebra (IT-College)

College Algebra (IT-College) Basic Algebraic Operations (Appendix A) Instructor: Eng. Eman Al.Swaity

Objectives Cover the topics in Appendix A (A3 ): A-3 : Polynomials : Factoring. Appendix A

A-3 : Polynomials : Factoring A-3 : Learning Objectives • After completing this Section, you should be able to: • Find the Greatest Common Factor (GCF) of a polynomial. • Factor out the GCF of a polynomial. • Factor a polynomial with four terms by grouping. • Factor a trinomial of the form . • Factor a trinomial of the form . • Indicate if a polynomial is a prime polynomial. • Factor a perfect square trinomial. • Factor a difference of squares. • Factor a sum or difference of cubes. • Apply the factoring strategy to factor a polynomial completely. A-3

A-3 : Polynomials : Factoring Introduction Factoring is to write an expression as a product of factors. For example, we can write 10 as (5)(2), where 5 and 2 are called factors of 10. We can also do this with polynomial expressions. In this section we are going to look at several ways to factor polynomial expressions. A-3

A-3 : Polynomials : Factoring Greatest Common Factor (GCF) The GCF for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. Factoring out the GCF Step 1: Identify the GCF of the polynomial. Step 2: Divide the GCF out of every term of the polynomial. This process is basically the reverse of the distributive property. A-3

A-3 : Polynomials : Factoring Example 1: Factor out the GCF: Step 1: Identify the GCF of the polynomial. The largest monomial that we can factor out of each term is 2y. Step 2: Divide the GCF out of every term of the polynomial. *Divide 2y out of every term of the poly. Note that if we multiply our answer out, we should get the original polynomial. In this case, it does check out. Factoring gives you another way to write the expression so it will be equivalent to the original problem. A-3

A-3 : Polynomials : Factoring Example 2: Factor out the GCF: This problem looks a little different, because now our GCF is a binomial. That is ok, we treat it in the same manner that we do when we have a monomial GCF. Note that this is not in factored form because of the plus sign we have before the 5 in the problem. To be in factored form, it must be written as a product of factors Step 1: Identify the GCF of the polynomial. This time it isn't a monomial but a binomial that we have in common. Our GCF is (3x -1). A-3

A-3 : Polynomials : Factoring Step 2: Divide the GCF out of every term of the polynomial. *Divide (3x - 1) out of both parts When we divide out the (3x - 1) out of the first term, we are left with x. When we divide it out of the second term, we are left with 5. That is how we get the (x + 5) for our second ( ). A-3

A-3 : Polynomials : Factoring Factoring a Polynomial with Four Terms by Grouping In some cases there is not a GCF for ALL the terms in a polynomial. If you have four terms with no GCF, then try factoring by grouping. Step 1: Group the first two terms together and then the last two terms together. Step 2: Factor out a GCF from each separate binomial. Step 3: Factor out the common binomial A-3

A-3 : Polynomials : Factoring Example 3: Factor by grouping: Note how there is not a GCF for ALL the terms. So let’s go ahead and factor this by grouping. Step 1: Group the first two terms together and then the last two terms together. Step 2: Factor out a GCF from each separate binomial. *Two groups of two terms *Factor out an x squared from the 1st ( ) *Factor out a 2 from the 2nd ( ) A-3

A-3 : Polynomials : Factoring Step 3: Factor out the common binomial. *Divide (x + 3) out of both parts Note that if we multiply our answer out, we do get the original polynomial. A-3

A-3 : Polynomials : Factoring Example 4: Factor by grouping: Note how there is not a GCF for ALL the terms. So let’s go ahead and factor this by grouping. Step 1: Group the first two terms together and then the last two terms together. Be careful. When the first term of the second group of two has a minus sign in front of it, you want to put the minus in front of the second ( ). When you do this you need to change the sign of BOTH terms of the second ( ) as shown above. *Two groups of two terms A-3

A-3 : Polynomials : Factoring Step 2: Factor out a GCF from each separate binomial. Step 3: Factor out the common binomial. Note that if we multiply our answer out that we do get the original polynomial. *Factor out a 7x squared from the 1st ( ) *Nothing to factor out from the 2nd ( ) *Divide (x - 2) out of both parts A-3

A-3 : Polynomials : Factoring Factoring Trinomials of the Form Basically, we are reversing the FOIL method to get our factored form. We are looking for two binomials that when you multiply them you get the given trinomial. Step 1: Set up a product of two ( ) where each will hold two terms. It will look like this: ( )( ). Step 2: Find the factors that go in the first positions. To get the x squared (which is the F in FOIL), we would have to have an x in the first positions in each ( ). So it would look like this: (x )(x ). A-3

A-3 : Polynomials : Factoring Step 3: Find the factors that go in the last positions. The factors that would go in the last position would have to be two expressions such that their product equals c (the constant) and at the same time their sum equals b (number in front of x term). As you are finding these factors, you have to consider the sign of the expressions: If c is positive, your factors are going to both have the same sign depending on b’s sign. If c is negative, your factors are going to have opposite signs depending on b’s sign. A-3

A-3 : Polynomials : Factoring Example 5: Factor the trinomial : Note that this trinomial does not have a GCF. So we go right into factoring the trinomial of the form Step 1: Set up a product of two ( ) where each will hold two terms. It will look like this: ( )( ) Step 2: Find the factors that go in the first positions. Since we have a squared as our first term, we will need the following: (a )(a ) A-3

A-3 : Polynomials : Factoring Step 3: Find the factors that go in the last positions. We need two numbers whose product is -14 and sum is -5. That would have to be -7 and 2. Putting that into our factors we get: Note that if we would multiply this out, we would get the original trinomial *-7 and 2 are two numbers whose prod. is -14 and sum is -5 A-3

A-3 : Polynomials : Factoring Example 6: Factor the trinomial : Note that this trinomial does have a GCF of 2y. We need to factor out the GCF before we tackle the trinomial part of this. We are not finished, we can still factor the trinomial. It is of the form . Anytime you are factoring, you need to make sure that you factor everything that is factorable. Sometimes you end up having to do several steps of factoring before you are done. *Factor out the GCF of 2y A-3

A-3 : Polynomials : Factoring Step 1 (trinomial): Set up a product of two ( ) where each will hold two terms. It will look like this: 2y( )( ) Step 2 (trinomial): Find the factors that go in the first positions. Since we have x squared as our first term, we will need the following: 2y(x )(x ) Step 3 (trinomial): Find the factors that go in the last positions. We need two numbers whose product is 15 and sum is 8. That would have to be 5 and 3. Putting that into our factors we get *5 and 3 are two numbers whose prod. is 15 *and sum is 8 A-3

A-3 : Polynomials : Factoring Factoring Trinomials of the Form (where a does not equal 1) Again, this is the reverse of the FOIL method. The difference between this trinomial and the one discussed above, is there is a number other than 1 in front of the x squared. This means, that not only do you need to find factors of c, but also a. Step 1: Set up a product of two ( ) where each will hold two terms. It will look like this ( )( )Step 2: Use trial and error to find the factors needed. The factors of a will go in the first terms of the binomials and the factors of c will go in the last terms of the binomials. The trick is to get the right combination of these factors. You can check this by applying the FOIL method. If your product comes out to be the trinomial you started with, you have the right combination of factors. If the product does not come out to be the given trinomial, then you need to try again. A-3

A-3 : Polynomials : Factoring Example 7: Factor the trinomial : Note that this trinomial does not have a GCF. So we go right into factoring the trinomial of the form Step 1: Set up a product of two ( ) where each will hold two terms. It will look like this: ( )( ) Step 2: Use trial and error to find the factors needed. In the first terms of the binomials, we need factors of 3 x squared. This would have to be 3x and x. In the second terms of the binomials, we need factors of 2. This would have to be 2 and 1. I used positives here because the middle term is positive. A-3

A-3 : Polynomials : Factoring Possible Factors Check using the FOIL Method First try: This is not our original polynomial. So we need to try again. Second try: This is our original polynomial. So this is the correct combination of factors for this polynomial. This process takes some practice. After a while you will get used to it and be able to come up with the right factor on the first try. A-3

A-3 : Polynomials : Factoring Factoring a Perfect Square Trinomial OR It has to be exactly in this form to use this rule. When you have a base being squared plus or minus twice the product of the two bases plus another base squared, it factors as the sum (or difference) of the bases being squared. This is the reverse of the binomial squared. Recall that factoring is the reverse of multiplication. A-3

A-3 : Polynomials : Factoring Example 8: Factor the perfect square trinomial: First note that there is no GCF to factor out of this polynomial. Since it is a trinomial, you can try factoring this by trial and error shown above. But if you can recognize that it fits the form of a perfect square trinomial, you can save yourself some time. Note that if we would multiply this out, we would get the original polynomial. *Fits the form of a perfect sq. trinomial *Factor as the sum of bases squared A-3

A-3 : Polynomials : Factoring Example 9: Factor the perfect square trinomial: First note that there is no GCF to factor out of this polynomial. Since it is a trinomial, you can try factoring this by trial and error shown above. But if you can recognize that it fits the form of a perfect square trinomial, you can save yourself some time. Note that if we would multiply this out, we would get the original polynomial. *Fits the form of a perfect sq. trinomial *Factor as the sum of bases squared A-3

A-3 : Polynomials : Factoring Factoring a Difference of Two Squares Note that the sum of two squares DOES NOT factor. Just like the perfect square trinomial, the difference of two squares has to be exactly in this form to use this rule. When you have the difference of two bases being squared, it factors as the product of the sum and difference of the bases that are being squared. This is the reverse of the product of the sum and difference of two terms found in Tutorial 6: Polynomials. Recall that factoring is the reverse of multiplication. A-3

A-3 : Polynomials : Factoring Example 11: Factor the difference of two squares: First note that there is no GCF to factor out of this polynomial. This fits the form of a the difference of two squares. So we will factor using that rule: Note that if we would multiply this out, we would get the original polynomial. *Fits the form of a diff. of two squares *Factor as the prod. of sum and diff. of bases A-3

A-3 : Polynomials : Factoring Example 12: Factor the difference of two squares: First note that there is no GCF to factor out of this polynomial. This fits the form of a the difference of two squares. So we will factor using that rule: Note that if we would multiply this out, we would get the original polynomial. *Fits the form of a diff. of two squares *Factor as the prod. of sum and diff. of bases A-3

A-3 : Polynomials : Factoring Factoring a Sum of Two Cubes The sum of two cubes has to be exactly in this form to use this rule. When you have the sum of two cubes, you have a product of a binomial and a trinomial. The binomial is the sum of the bases that are being cubed. The trinomial is the first base squared, the second term is the opposite of the product of the two bases found, and the third term is the second base squared. A-3

A-3 : Polynomials : Factoring Example 13: Factor the sum of cubes: First note that there is no GCF to factor out of this polynomial. This fits the form of the sum of cubes. So we will factor using that rule: Note that if we would multiply this out, we would get the original polynomial. *Fits the form of a sum of two cubes *Binomial is sum of bases *Trinomial is 1st base squared, minus prod. of bases, plus 2nd base squared A-3

A-3 : Polynomials : Factoring Factoring a Difference of Two Cubes This is factored in a similar fashion to the sum of two cubes. Note the only difference is that the sign in the binomial is a - which matches the original sign, and the sign in front of ax is positive, which is the opposite sign. The difference of two cubes has to be exactly in this form to use this rule. When you have the difference of two cubes, you have a product of a binomial and a trinomial. The binomial is the difference of the bases that are being cubed. The trinomial is the first base squared, the second term is the opposite of the product of the two bases found, and the third term is the second base squared. A-3

A-3 : Polynomials : Factoring Example 14: Factor the difference of cubes: First note that there is no GCF to factor out of this polynomial. This fits the form of the difference of cubes. So we will factor using that rule: Note that if we would multiply this out, we would get the original polynomial. *Fits the form of a diff. of two cubes *Binomial is diff. of bases *Trinomial is 1st base squared, plus prod. of bases, plus 2nd base squared A-3

A-3 : Polynomials : Factoring Factoring Strategy I. GCF: Always check for the GCF first, no matter what. II. Binomials: a. b. c. A-3

A-3 : Polynomials : Factoring Factoring Strategy III. Trinomials: a. b. Trial and error: c. Perfect square trinomial: IV. Polynomials with four terms: Factor by grouping A-3

A-3 : Polynomials : Factoring Example 15: Factor completely: The first thing that we always check when we are factoring is WHAT? The GCF. In this case, there is one. Factoring out the GCF of 3 we get: Next, we assess to see if there is anything else that we can factor. We have a trinomial inside the ( ). It fits the form of a perfect square trinomial, so we will factor it accordingly: *Factor a 3 out of every term *Fits the form of a perfect sq. trinomial *Factor as the sum of bases squared A-3

A-3 : Polynomials : Factoring Example 16: Factor completely: *Fits the form of a diff. of two squares *Factor as the prod. of sum and diff. of bases *Fits the form of a diff. of two squares *Factor as the prod. of sum and diff. of bases A-3

A-3 : Polynomials : Factoring Example 17: Factor completely: *Factor by trial and error A-3

A-3 : Polynomials : Factoring Example 18: Factor completely: *Fits the form of a sum of two cubes *Binomial is sum of bases *Trinomial is 1st base squared, minus prod. of bases, plus 2nd base squared A-3

A-3 : Polynomials : Factoring Example 19: Factor completely: *Group in two's *Factor out the GCF out of each separate ( ) *Factor out the GCF of (x + 5b) A-3

End of the Lecture Let Learning Continue

C ollege A lgebra (IT-College)