Based on Faith Academy Posted February 25, 2015 Share Posted February 25, 2015 I have to understand why things are the way they are in a problem. I am stumped as to the why this problem and subsequent problems are the way they are. Solve and give every solution: They are third degree problems. An example would be x(cubed)minus 3x(squared)minus 6x plus eight equals zero. -2 is a solution. My failure to understand lies in the division part of the problem. What doesn't make sense to me is how is it that I can change the subtractions to additions and change the signs and get a completely different answer than if I just left the subtraction as subtraction and went with the knowledge that a subtraction automatically changes the sign to the opposite. Am I making sense? TIA Quote Link to comment Share on other sites More sharing options...
regentrude Posted February 25, 2015 Share Posted February 25, 2015 Solve and give every solution: They are third degree problems. An example would be x(cubed)minus 3x(squared)minus 6x plus eight equals zero. -2 is a solution. My failure to understand lies in the division part of the problem. What doesn't make sense to me is how is it that I can change the subtractions to additions and change the signs and get a completely different answer than if I just left the subtraction as subtraction and went with the knowledge that a subtraction automatically changes the sign to the opposite. Am I making sense? TIA No, you're not making sense to me. Here is how to do the polynomial division: Write out the problem: x^3-3x^2-6x+8=0. Guess x=-2 is solution. Divide by (x-(2)) : x^3-3x^2-6x+8 / (x+2)= ? we know leading term of ? must be x^2, so, multiply (x+2) by x^2: x^3+2x^2 This must be subtracted from original polynomial, just like regular long division. Are the signs in the next step where the confusion arises? x^3-3x^2-6x+8 - (x^3+2x^2) = - 5x^2-6x+8 (*) now we see next term of ? has to be - 5x multiply (x+2) by 5x: - 5x^2 -10x this must be subtracted from remaining polynomial (*) - 5x^2-6x+8 (*) - (- 5x^2 -10x) = 4x+8 so last term of ? has to be 4 (x+2)*4=4x+8, when this is subtracted there remains no rest. Thus, x^3-3x^2-6x+8 / (x+2)= x^2 - 5x +4 Tell me where exactly you have the problem, so I can help. 1 Quote Link to comment Share on other sites More sharing options...
Based on Faith Academy Posted February 25, 2015 Author Share Posted February 25, 2015 This must be subtracted from original polynomial, just like regular long division. Are the signs in the next step where the confusion arises? x^3-3x^2-6x+8 - (x^3+2x^2) = - 5x^2-6x+8 (*) This is the start of what makes no sense of why it is so. In TT they changed the subtraction signs in the original problem to addition and made the problem read....x^3+-3x^2+-6x+8. If you add doing the problem that way you get what you got BUT if you leave the signs as subtraction then you would get -x^2- 6x+8. This makes no sense to me. Whether you change the sign at the start of the problem or as you go through the problem, you should still get the same answers(to my way of thinking) :huh: Quote Link to comment Share on other sites More sharing options...
regentrude Posted February 25, 2015 Share Posted February 25, 2015 This is the start of what makes no sense of why it is so. In TT they changed the subtraction signs in the original problem to addition and made the problem read....x^3+-3x^2+-6x+8. If you add doing the problem that way you get what you got BUT if you leave the signs as subtraction then you would get -x^2- 6x+8. This makes no sense to me. Whether you change the sign at the start of the problem or as you go through the problem, you should still get the same answers(to my way of thinking) :huh: x^3+-3x^2+-6x+8. is exactly the same as x^3 -3x^2 - 6x+8. There is absolutely no difference between the two expressions, so no, I do not understand what you are saying. You have to subtract : .x^3+-3x^2+-6x+8. - (x^3+2x^2) The x^3 terms cancel - 3x^2 - 2x^2 = -5x^2, irrespective of whether you write an extra "+" in front of the -3x^2 Quote Link to comment Share on other sites More sharing options...
Based on Faith Academy Posted February 25, 2015 Author Share Posted February 25, 2015 x^3+-3x^2+-6x+8. is exactly the same as x^3 -3x^2 - 6x+8. There is absolutely no difference between the two expressions, so no, I do not understand what you are saying. Exactly!! You have to subtract : .x^3+-3x^2+-6x+8.OK but if you wrote it as x^3-3x^2-6x+8 then.... - (x^3+2x^2)This is my problem.. When you subtract x^3-2x^2.... The x^3 terms cancel - 3x^2 - 2x^2 = -5x^2, irrespective of whether you write an extra "+" in front of the -3x^2...Then -3x^2-(-2x^2)=(-x^2)?? Shouldn't the -2x^2 change to positive? Quote Link to comment Share on other sites More sharing options...
kiana Posted February 25, 2015 Share Posted February 25, 2015 You shouldn't be subtracting (x^3 - 2x^2). You should be subtracting (x^3 + 2x^2). Why do you feel the sign on the 2 should change? Sometimes people are taught to add the negative instead -- so they would be adding (-x^3 - 2x^2) -- this is mathematically legitimate even though I don't care for it because I feel it confuses people more. Have you gotten this mixed up with the subtracting somehow? 2 Quote Link to comment Share on other sites More sharing options...
letsplaymath Posted February 26, 2015 Share Posted February 26, 2015 Are you trying to divide by (x - 2) instead of (x + 2)? That was a typo up above, I think. If x = -2 is a solution, then the factor you need to divide out is x - (-2), which means x+2. That's because if x = -2 is a solution, then the function has to = 0 when x is -2. And that means the function must have a factor that would turn into zero at that point. The factor that becomes zero when x = -2 is (x + 2), since -2 + 2 = 0. 2 Quote Link to comment Share on other sites More sharing options...
Based on Faith Academy Posted February 26, 2015 Author Share Posted February 26, 2015 Are you trying to divide by (x - 2) instead of (x + 2)? That was a typo up above, I think. If x = -2 is a solution, then the factor you need to divide out is x - (-2), which means x+2. That's because if x = -2 is a solution, then the function has to = 0 when x is -2. And that means the function must have a factor that would turn into zero at that point. The factor that becomes zero when x = -2 is (x + 2), since -2 + 2 = 0. After you divide into x^3 you get x^2 and multiply x^2 by x + 2. From the way you showed the problem, you put an addition sign between x^3 + 2x^2. I thought we had to bring down the subtraction sign that was in the problem(x^3-3x^2). If I brought down the subtraction sign then the answer would have been different then if I put an addition sign in. Is my thinking wrong here? Quote Link to comment Share on other sites More sharing options...
kiana Posted February 26, 2015 Share Posted February 26, 2015 After you divide into x^3 you get x^2 and multiply x^2 by x + 2. From the way you showed the problem, you put an addition sign between x^3 + 2x^2. I thought we had to bring down the subtraction sign that was in the problem(x^3-3x^2). If I brought down the subtraction sign then the answer would have been different then if I put an addition sign in. Is my thinking wrong here? Yes. You are not bringing down the subtraction sign from the original problem. You're subtracting off the result of x^2 (x + 2). If your original problem began with x^3 + 3x^2 your x^2 term would be (3x^2 - 2x^2). 1 Quote Link to comment Share on other sites More sharing options...
Based on Faith Academy Posted February 26, 2015 Author Share Posted February 26, 2015 I've got it!!! My problem was that I was focused on the problem and not on the signs of the divisor when I multiplied. That was what was determining whether the problem had a subtraction or addition between x^3 and 2x^2.Thank you so much for all your help!!!! I kept looking at it and finally realized the answer was right there all along. 2 Quote Link to comment Share on other sites More sharing options...
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