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Dividing polynomials by linear expressions

Learn how to divide polynomials by linear factors! This tutorial guides you through the process of algebraic long division, focusing on highest degree terms first. You'll master the art of subtracting polynomials, handling remainders, and rewriting expressions in different forms.

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  • cacteye yellow style avatar for user Br Paul
    Is dividing the polynomials by algebraic long division the only way to solve this problem?
    (16 votes)
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  • duskpin sapling style avatar for user Zoe Gesmundo
    in why does 2x^2 become 2x
    (5 votes)
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    • female robot grace style avatar for user loumast17
      The steps of polynomial long division are as follows.

      1) find the term you have to multiply the leading term of the divisor (denominator) you have to multiply by to get the first term of the dividend (numerator.) In this case the denominator is x+2 and the numerator is 3x^3 + 4x^2 -3x +7. We want what we have to multiply x in x+2 by to get 3x^3. so x times what is 3x^3? the answer is 3x^2. This is the first term of your answer (the quotient.) And just to make sure other terms are explicitly stated, the dividend is 3x^3 + 4x^2 -3x +7 and divisor is x + 2

      2) take the term of the quotient you just found and multiply it by the divisor. so the term found was 3x^2 and the divisor is x + 2, so ultiply those to get 3x^3 + 6x^2. You should notice 3x^3 is the same as the first term in the dividend.

      3) subtract this new expression from the dividend. so (3x^3 + 4x^2 -3x +7) - (3x^3 + 6x^2) = -2x^2 - 3x + 7. THIS is your new divisor, or in other words you can think of the division problem now as (-2x^2 - 3x + 7) / (x + 2).

      4) Repeat steps 1-3 to find the next term of your quotient, the expression you subtract from the current dividend and then the new dividend.

      so step 1, what do we multiply the first term of the divisor by to get the first term of the dividend? -2x (this is where the -2x you asked about came from.) So the second term of the quotient is -2x, Then you multiply by the dividend -2x(x + 2) = -2x^2 - 4x. now subtract this from the current dividend (-2x^2 - 3x + 7) - (-2x^2 - 4x) = x + 7.

      Now just keep going until the dividend has a smaller degree than the divisor. Then this is your remainder.

      Let me know if that didn't help. just to make sure your question specifically is answered, the -2x is the number that multiplies the first term of the divisor to get the first term of the new dividend, which is part of the process of solving these problems.
      (17 votes)
  • stelly yellow style avatar for user Tony Punnacherry
    What happened to the synthetic division video and practice?
    (9 votes)
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  • blobby green style avatar for user Tyler Wood
    Do we still have to put a condition on the domain that 'x cannot equal -2', even though the remainder of 'x + 2' clearly shows that a value of -2 would create an undefined denominator?
    Asking for a friend...
    (6 votes)
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    • leaf orange style avatar for user A/V
      There are many ways to write the domain, and it really depends on your teacher. For the easiest, you may write (just going off you're example), (-inf,inf) x ≠ -2.

      More formally, which would be nice, is to use the "union term." More specifically, instead of (-inf,inf) x ≠ -2, it would be (-inf,-2]U[-2,inf). Why union? It's the term for the combination of two unconnected intervals. Hopefully this helps.
      (3 votes)
  • male robot hal style avatar for user chrisliao101
    I have a question. When Khan does 4 - (-6) he gets 2. However, in practice, I have to make the 6 positive and get 10. Why doesn't Khan fix the problem?
    (3 votes)
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  • leaf blue style avatar for user Visontai Barnabás
    In Hungary, we do not learn (algebraic) long division, therefore I do not understand what Sal is doing in the video. As far as I am concerned, dividing polynomials by linear expressions is needed for some integration problems, so could you recommend me a video in connection with the topic, or suggest another method?
    (2 votes)
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  • blobby green style avatar for user 100049
    in the past video we would stop and end it as a remainder in ,, why are we continuing? or the question is when do I know to stop
    (2 votes)
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  • starky seedling style avatar for user maycroft,toby
    i no understand
    (2 votes)
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  • aqualine ultimate style avatar for user Simum
    Why do we always focus on the highest degree term? Can't we start dividing by the constant?
    (2 votes)
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  • blobby green style avatar for user Arghya Das
    We Know Dividend = Divisor * Quotient + Reminder
    So 3x^3+4x^2-3x+7 Should Be Written As (x+2)(3x^2-2x+1)+5,
    Buy Why It Is Written As 3x^2-2x+1+(5/x+2)?
    (2 votes)
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    • leaf orange style avatar for user A/V
      First things first, both expressions you wrote are the same thing, formatted differently. At the end, it is a stylistic choice, but the preferred way is the fraction remainder.

      If we perform arithmetic division on two numbers, let's say 12/5, our remainder would not be 2; rather, it'd be:
      (2/5) — remainder/divisor, and we would write it as
      (12/5) = 2 + (2/5)

      Same line of reasoning with polynomial division !
      hopefully that helps !
      (1 vote)

Video transcript

- We're told, divide the polynomials. The form of your answer should either be just a clean polynomial, or some polynomial plus some constant over x plus two, where p of x is a polynomial and k is an integer. Fair enough, and if we were doing this on Khan Academy, this is a screen shot from Khan Academy, we would have to type this in, but we're just going to do it by hand. And, like always, pause this video and try to do it on your own before we work through it together. All right, now let's work through it together. And what we're trying to do is divide x plus two, into three x to the third power plus four x squared minus three x plus seven. And so, like always, we focus on the highest degree terms first. x goes into three x to the third power how many times? Three x squared times. We'd want to put that in the second degree column. Three x squared. Three x squared times two is six x squared. Three x squared times x is three x to the third power. There's something very meditative about algebraic long division. Anyway, we'd want to subtract what we just wrote from what we have up here. So let's subtract. And these characters cancel out. And then four x squared minus six x squared is negative two x squared. Bring down that negative three x. And now we would wanna say, hey, how many times does x go into negative two x squared? Well, it would go negative two x times. Put that in our first degree column. Negative two x times two is negative four x. Negative two x times x is negative two x squared. Now we wanna subtract what we have here in orange from what we have up here in teal. So we either put a negative around the whole thing or we distribute that negative and that becomes a positive, that becomes a positive. And so this is equal to, the x squared terms cancel out. Negative three x plus four x is just going to be a straight up x. Bring down that seven, x plus seven. How many times does x go into x? Well, one time. Actually, let me do a, use a new color here. So, how many times does x go into x? It goes one time. Put that in the constant column. One times two is two. One times x is x. We wanna subtract these characters. And we're left with seven minus two is five. And so, we can rewrite this whole thing as, we deserve I guess a little bit of a drum roll. Three x squared minus two x plus one, plus the remainder, five, over x plus two. One way to think about it is, hey, I have this remainder, I'd have to keep dividing it by x plus two if I really wanted to figure out exactly what this is. Now if I wanted these expressions to be completely identical I would put a condition on the domain that x cannot be equal to negative two because if x was equal to negative two, we'd be dividing by zero here. But for the purposes of this exercise, you just have to input this part right over here, you'd have to type it in, which I guess isn't the easiest thing to do in the world. But it's worth doing. All right, see you in the next video.