You may think that figuring out how to factor this into the product of two binomials is an awful chore. But these problems are actually quite easy when you use a system to unfoil them. You go through the system and it helps you find what the answer is or to determine if there isn’t an answer. This isn’t the case with all the factoring problems but it is true of quadratics in the form AX² +BX+C.
That is why quadratics are so nice to work with an algebra. The key to unfoiling these factoring problems is organization. Be sure you have an expression in the form AX² + BX + C. Be sure the terms are written in the order of decreasing powers, if needed review the list of prime numbers and perfect squares.
And finally follow the steps, try this example which is written in the order of decreasing powers using unfoil. First determine all the ways you can multiply two numbers to get A, every number can be written as at least one product. Even if it is only the number times one, so assume that there are two numbers E and F – whose product is equal to A. These are the two numbers you want for this problem.
Second, determine all the ways you can multiply two numbers together to get C. If the value of C is negative, ignore the negative sign for the moment. Concentrate on what factors results in the absolute value of C. Now assume that there are two numbers G and H, whose product is equal to C, use these two numbers for this problem. Third, look at the sign of C and your list from steps one and two, if C is positive find the value from your step one list. And another from your step two list such that the sum of their product and the product of the two remaining numbers in those steps results in B.
If C is negative, find the value from your step one list and another from your step two list. Such that the difference of their product and the product of the two remaining numbers from those steps results in B. Fourth, arrange your choices as binomials, the E and F have to be in the first positions in the binomials. And the G and H have to be in the last positions, they have to be arranged so the multiplications in step three have the correct outer and inner products.
Fifth, place the signs appropriately the signs are both positive if C is positive and B as positive. The sign are both negative if C is positive and B is negative, one sign is positive and one negative. If C negative, the choice depends on whether be as positive or negative and how you arrange the factors. Now let’s try it out, using unfoil follow these steps to factor the quadratic 2X² - 5X – 12, which is in the form AX² + BX + C and written in the order of decreasing powers. First determine all the ways you can multiply two numbers to get A.
You can get these numbers from the prime factorization of A. Prime numbers can only be divided by themselves and one. Here are some examples, prime factorization is finding the prime numbers that divide any given value. Sometimes writing out the list of ways to multiply is a big help. Here is one example, in the example 2X² - 5X – 12, the value of A is 2. The only w ay to multiply two numbers together to get two is 1(2), second determine all the ways you can multiply two numbers to get C.
Remember in this example the value of C is -12, ignore the negative sign right now. The negative becomes important in the next step, just concentrate on what multiplies together to give you 12. There are three ways to multiply two numbers together to get 12, remember to look at the sign of C and your list from steps one and two. The third step is to choose a product from step one and a product from step two. In this example, C is -12 and B is 5. So look for a combination from step one and step two who’s different results in five.
Use the 1(2) from step one and the 3(4) from step two. Multiply the one from step one, times the three from step two. And then multiply the two from step one times the four from step two. The two products are three and eight, whose difference is five, the fourth step is to arrange your choices as binomials. So the results are those you want, from the example the following arrangement multiplies the 1X by the 2X. To get the 2X² which is needed for the first product, likewise the four the three multiply to give you 12.
The outer product is 3X and the inner product is 8X, finally place the signs to give the desired results. This next example looks at first like a great candidate for factoring by this method. But you will see that not everything can factor, also using this method assures that you have left no stone unturned. First, determine all the ways you can multiply two numbers to get A. Second, determine all the ways you can multiply two numbers to get C, third – look at the sign of C and your list from steps one the two to see if you want a sum or a difference.
The last term is negative so you want the difference of the products to be 27. Fourth, choose a product from step one and a product from step two, you can’t seem to find any combination that gives you a difference of 27. Run through to be sure that you haven’t miss anything. Using the 1(18) crossing it with 1(4) gives you a difference of either 14 using the 1(4) and 18(1) or 71 using the 1(1) and the 18(4). Using 2(2) gives you a difference of 34 using 1(2) and 18(2), there is only one choice because both of the second factors are two.
Do the math crossing 2(9) with 1(4) or 2(2) and you still do not get the desired result. Use 3(6) and then cross it with 1(4) and 2(2) and it won’t be any better. Because you exhausted all the possibilities and haven’t been able to create a difference of 27, you can assume that this quadratic can not be factored. It is prime.
Transcription by:
Scribe4you Transcription Services