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Factor $x^{2}-x-12$ over the integers....

Question

Factor $x^{2}-x-12$ over the integers.

Factor $x^{2}-x-12$ over the integers.



Answers

Factor by grouping.
$$x^{3}-3 x^{2}+4 x-12$$

Look for great is coming. Factor noticed. Each term has an X minus one in it. So we're back to our X minus one and we're left with 12 X square minus four X minus five. From here, we want a factor. The trying no meal. We take 12 in negative five. Multiply them. You get negative. 60 negative sixties air target number as our product. And we got to come up with factors of negative 60 ads. Negative four. So you could try different combinations like, um, 10 in six. If the 10 was negative and the six was positive, they would add to negative four multiplied and negative 60. So that's our combination. So what we do with this is we break apart the middle term. We're un combining like terms were turning that negative for axe into a negative 10 x plus six x So we're turning the trying Amiel into four terms. Now we can group use the grouping method in greatest common factor. Each of these groups we could take out a two X left with six x minus five and then we could just take out a positive one and left with six x minus five so that with these two terms, we have the common various common factor of six X minus five. Then when we divide that I was left with two x plus one. And don't forget about the X minus one that we accurate out at the very beginning. That's got to be part of this as well. So in no particular order with these groups, you should just have three of them. An x minus one a six x minutes by

We need to find the factors off. 66 square minus 40 next minus 12 1st of all, we will divide the whole expression by two. So we're left with three X Square minus seven X minus six. Now we'll split the middle tum into two numbers and be such that the some off a plus B is equal to minus seven. Noted the coefficient of X on the product off a n B is equal to the product off the coefficient of X square and a constant, um, that is minus 18. So the two numbers would be minus nine and two, so we can write three X Square minus nine X Blessed to X minus six. Now we get dicked three ex common from these totems So three X times X minus tree. Now we can take to common from these two domes. So two ex my mystery No, we can dig X minus three common so extra Ministry times three X plus two No de stumps committed and as X minus tree times expressed to divide by three D are required factors

We're looking for two numbers that I'm too negative. 13 and multiply two. Positive 12. So now let's see what numbers add to negative 13 and bolts by 12. Well, first off, multiplying to 12. We know we have six times, too, but six plus two isn't negative. 13. We know we have four times three is 12 before Post three isn't nine of 13. We also know we have 12 times one. Well, 12 plus one is positive. 13. What about negative 12? Minus one? Well, that's negative. 13. So what this indicates is we would have access minus one times X minus 12. We put negative signs on each of the numbers because we know that we're adding up to a negative value.

First thing we're going to do, it's to rearrange our function of X squared plus X is equal to 12 to match the formula of a X squared plus B X plus c. To do that, we will subtract 12 from each side to get X squared. Plus X minus 12 is equal to zero. Now that we have this we can use. The formula of X is equal to negative B plus minus square root of B squared, minus four a c over two A. Our values of a is one be is one and C is negative. 12. Putting our values together. You will get negative one plus minus square root of one squared minus four times one multiply by negative 12 All of this is over two times one. So first we are going to do the plus. This will look like negative one plus swear route of one squared when it's four times one multiplied by negative 12 all of that over two times. One. To simplify, we will get negative one plus one minus four times negative 12 over. To simplified further, you will get negative one plus one plus 48 over, to which is negative one plus 49 square root of 49 over to and we know that square root of 49 busy cool to seven. So negative one plus seven over to final simplification. We get six over to which is equal to three. Now we're going to do the minus, which is negative one minus square root of one squared minus four times one times negative. 12 All of that over. Two times one simplified. We will get negative one minus one minus 48 or plus 48 over to. And we will get negative one minus swear route of 49 over. To simplified further, we will get negative one minus 7/2, which is equal to negative eight over to which is also negative for So for this problem, our X is equal to three and negative four.


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