Factoring: “ax 2 +bx+….wha?”

1. Factoring:“ax2+bx+….wha?” By Gaelan Bandong

2. Having Trouble? • Oh ho! I see your having trouble with factoring, well, don’t worry! I’m here to help, and I will try to make this as easy as possible!

3. What's Factoring? • Its important to know what and why factoring was introduced. First of all, it was introduced as a means to exercise the minds of children, and it is just an obstacle that people hope our children will be able to pass…..and you need it to pass the SATS.

4. What's Factoring? (cont) • Factoring, in terms of math, is a way to separate numbers or polynomials (ex. x2+5x+6 factors into (x+3)(x+2).) into two or more terms. • Don’t worry if you are thinking that this is some foreign language like, 一に産し語六世界, its not really hard and I will be helping you with the following presentation. • My best advice I can give you is, DON’T PANIC!!

5. Examples of multiplying • Monomial x binomial: a(b+c) use Distributive property to get ab+ac • Monomial x trinomial: a(ax2+bx+c) look up • Binomial x binomial: (ax+b)(ax+c) use F.O.I.L.to get ax2+bx+c • Binomial x trinomial: (ax+b)(ax²+bx+c) use Box method to get a²x3+2abx+b²x+ac+bc (not always that long so don’t freak out!) • Trinomial x trinomial: (ax²+bx+c)(ax²+bx+c) • All underlined terms are in the glossary.

6. Factoring cross • When trying to factor Trinomials like ax²+bx+c, you have to use a factoring cross where you put a multiplied by c on the top, and b on the bottom. On the left and right you put two numbers that multiply into the top, and add into the bottom number.

7. Common factors • Common factors are well, just that. Say you have 300x²+300b+300 now your first thought is probably THAT’S TO BIG TO DO A FACTOR CROSS!!!!!!!!!!!!!!!!, its really simple what you have to do. Since 300 is a factor for all of them you can factor it out to make 300(x²+x+1) which is not so hard.

8. Two terms: Difference of squares • A difference of squares is when you have a binomial that looks like say, 4x²-49, you should notice that both have a square root, which is 2 and 7 respectively. With this you can factor it to make it into (2x+7)(2x-7) right away. Don’t forget the NEGATIVE, because if you do, 100% it will be wrong and it wouldn’t be a difference of squares. • Note: only works on binomials with a subtraction sign and both numbers have a square root.

9. Three terms: Trinomial squares • Trinomial squares are the result of binomial squares which is a binomial that looks like this (x+b)² or (x-b)² where the result is either x²+2bx+b² or x²-2bx+b². You can tell if it is a trinomial square if the first number has a square root, the second number can be divided by 2 to get two of the same numbers and the last number also has a square root. This type of trinomial can be a relatively hard one to spot or an easy one depending how closely you look at trinomials.

10. Trinomial: a=1 • When you have a trinomial where the “a” value is equal to 1 such as x²+5x+6*. Since this is not a trinomial square, then you have to use the factor cross. After using the factor cross you should get the factors of 2 and 3 ( 2 and 3 add up to 5, which should be on the bottom, and multiply into 6 which is on the top). Since a=1 you don’t have to do anything special and you’ll get (x+2)(x+3) • *Since there is a 1 in front of the x you don’t need to show the 1.

11. Trinomial:a=1 • When you have a trinomial like this then you would have to do something slightly different. You still have to do the cross, but this time you have to multiply a and c together. After you do this you have one extra step to do, which is to box factor it, it is essentially the same as the regular box method, but backwards. The first term goes into the first box (top left), the two terms that you got from the cross go into the second (top right) and the third (bottom left) and the last term in the trinomial will be in the last box (bottom right). Remember that the two terms you got will always have an x after them. • If you still don’t get it, there is an example of the box and reverse box method in the glossary.

12. Four terms: Grouping • If you have four terms, you can factor them by grouping. Say you have 4x²+2x+8x²+4x, first you will have to turn it into two different groups. Now it should look like this (4x²+2x)+(8x3+4x²). If you notice, there are common factors in both groups. These are 2x and 4x respectively. After you factor it you should get 2x(2x+1)+4x²(2x+1). Notice that the two terms in parentheses are the same, coincidence? To finish off you should put 2x and 4x² together to get (2x+4x²) and since there are two (2x+1)s, you only use one and you should get (2x+4x²)(2x+1).

13. Factoring Completely • Say you have a problem like 16x4+1* and you factored it out to get (4x²+1)(4x²-1), but HAH! You’ve activated my trap card, its actually still not done (heh, I know that in your head you are like WOAHHHHH!). Notice that the second term is still a difference of squares and can be factored further to get (4x²+1)(2x+1)(2x-1). • *1 has a square root of 1 (1 x 1=1)

14. Future uses of Factoring • Later, you will have to use factoring to solve for 0. ex (5x+1)(x-7)=0. to solve this you would first translate each term into a separate equation (5x+1=0, and x-7=0) as the rules for equation solving you would have to subtract 1 from both sides for the first equation and add 7 to both sides to the second one. And then you would have to divide by 5 from both sides on the first one. The result should be x=1/5 and 7 which is your answer • The second use of factoring is almost like the first, but you have one more step. Say there’s a problem like (x+1)(x-1)=8, after you multiply (x+1)(x-1) you would get x²-1=8, then you would subtract 8 from both sides to get x²-1-8=0, 8 and 1 being like terms you will get x²-9=0. Now you have to factor this to get (x-3)(x+3)=0. After that its just like factoring to solve for 0.

15. Summary • All in all, factoring can be a long and tedious process, but can be fairly easy if you know what to do. I know there are many different types of factoring such as difference of squares, trinomial squares, and such, but each has its own, fairly simple to solve. There are two things that you need to know to keep you from freaking out about factoring that you need to know. 1. Don’t panic and you should be fine and 2. Its for passing the S.A.T. test.

16. Glossary • Factoring: look at 4th slide titled What’s factoring? • Monomial: One term or number ex. 3x • Binomial: Two terms or numbers ex. 3x²+6x • Trinomial: Three terms or numbers….big surprise. Ex.3x²+6x+9 • Factor Cross: The cross used to factor a trinomial. • Box method: The box used for multiplying a binomial/trinomial by a binomial/trinomial. Can also be used for factoring a trinomial Looks like this. • F.O.I.L: stands for first, outside, inside, last. In which you multiply the first terms first, then the outside terms (the first and the last), then the inside (second and third terms) and then you multiply the last two terms.

17. About the author • Hey, my names Gaelan Bandong (no its not spelled like it sounds). I’m 14 years old and I like to draw, play video games, watch movies etc. I also like to help people if they are having trouble, so if you have any problems just email me at Gaelan.Bandong@email.com if you want to see my art you can visit my art page here at http://issa-kun.deviantart.com/. Im also very good with computers, but there are those times where I want to punch through it.

18. Practice problems • Factor 1.2a²+6a 2.x²-4 3.x²+8x+15 4.2x²-7x-4 5.2x²-4x+xz-2z