1. What is the greatest common factor?
2. How do you know when you have found the greatest one?
1. Explain how to factor the following trinomials forms:
a. x2 + bx + c
b. ax2 + bx + c.
2. Is there more than one way to factor this? Show your answer using both words and mathematical notation.
- The greatest common factor (GCF) in a polynomial is the common factor with the greatest (integer) coefficient and highest degree. In my common language, it's the highest number that will go into all the expressions of a polynomial. The example from the text is 12b^3+8b^2 = 4b^2(3b+2). 4b^2 is the greatest combination of integer (4) which can be multiplied to 12 and 4, the highest degree power of variable that can be multiplied to equal b^3 and b^2 is b^2.
2- You can check to made sure you have found the highest one by completely factoring each term of the polynomial into prime numbers and writing the powers as repeating multiplication. 12= 2*2*3=4*3 8=2*2*2=4*2 - The number 4 is the greatest integer that 12 and 8 have in common. b^3 = b*b*b, b^2=b*b, the highest common power is ^2.
Once done factoring the work for this equation [12b^3+8b^2 = 4b^2(3b+2)] can be checked with multiplication by using the distributive property to the right side of the equation . In this case 4b^2(3b+2) 4b^2*3b=12b^3 + 4b^2*2=8b^2 12b^3+8b^2 It checks, it is correct.
I agree with you I too feel pressure when working on math problems. Especially Algebra I had my first algebra class the last nine weeks and I can not tell you how many times I just felt like giving up but I kept at it. I didn't really understand some of it but I still tried to figure it out. I too need to start taking time to check and recheck my problems to make sure I understand and get the right answer. Any ideas?
Thanks for your help to everyone on this! :-)
Many students simply use trial and error as their main process. The only way to really manage the trial and error is to keep practicing. With more practice that you get under your belt and the more ways you run through each problem, then you will start to develop an "eye" for the factoring.
When we work with it a lot, we can glance at a problem and see the factors. Unfortunately, we get rusty. <sigh>
For me, I can usually look at a problem and figure out what needs to be factored out but I always make sure to work out each problem just in case. It is easy to make mistakes so I always try and double-check my work. In math, if you are one number off you will throw off the whole problem and get an incorrect answer so you should always try and play it safe.
To factor x² + bx + c, you first have to find two numbers that are the product of c and the same two numbers have to be the sum of b. An example is x² + 5x + 4, the two numbers here would be 4 and 1. Because 4 times 1 is 4, and 4 plus 1 is 5. Factoring the trinomial is, x² + 5x + 4 = (x+4)(x+1).
To factor ax² + bx + c, the first thing is to factor out the greatest common factor. An example is 2x² + 6x + 4, is rewritten as 2(x² + 3x + 2) because the greatest common factor is 2. Then we find the 2 numbers that have a product of 2 and a sum of 3. Which would be 2 and 1, because 2 times 1 is 2, and 2 plus 1 is 3. To factor, it is written as 2(x + 2)(x + 1).
Your post did not give reasoning to why you said it could not be factored. You failed to give any visualization or explaining to your reasoning. Looking at your post I was a little confused on what you were trying to state. I am not saying you are wrong because I did not see what you were solving. I just wish you would have added a little more to what you were trying to say. I also am not sure to what problem this is related to. I think you may have an understanding to what steps would be taken to solve these problems. I also think you did a great job in trying to express your meaning. Just wish I could have seen a little more. Have a great Halloween.
The first question x2 + bx + c, the first step is to find the greatest common factor. In the case of this polynomial there is no greatest common factor. I would next try to factor by grouping. The process of foil is used to un-factor or solve for a variable. An example of this is (x^2) + 5x +4. This can be factored by (x+4)(X+1), solving for this results in x = -1,-4. I think the correct way to factor a is x(x+b)+c. The second question ax2 + bx + c, I believe is done the same as the first question, the only difference is the a in from of the x squared term. x(ax + b) + c. I think that this is the only way to factor this equation. X is the greatest common factor in the first two terms. The last term c is not in any of the first two terms. This is why I factored out x. I don't see how the equation could be factored any other way.
Sometimes it can take a while for people to find the greatest common factor, because some students simply do it the long way which may seem easier to them. Finding the greatest common factor is easy and quick when you use prime factoring.
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