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Course: Algebra (all content)>Unit 10

Lesson 23: Practice dividing polynomials with remainders

Dividing polynomials with remainders

Sal divides (x^3+5x-4) by (x^2-x+1) using long division. Created by Sal Khan and Monterey Institute for Technology and Education.

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• OK, so my teacher gave us a problem just like this. But one of the terms comes out as a zero! what do i do??? :(
• You can write the 0x^...If you want to but you need to remember you're either adding or subtracting a number from 0, which is the just opposite of whatever number you have...
• Can't you factor a 5 out of 5x - 5 at ? Also, you could factor x^2 - x +1 to get (x - 1)(x - 1). So, you would have 5(x - 1) divided by (x - 1)(x - 1). This would result in 5 divided by x - 1. So, shouldn't the final answer be x + 1 + 5/x-1?
• How do you handle a polynomial division when the initial terms don't divide evenly? For example, 2x - 5 divided into 3x^3 -9x^2 +15. Thanks!
• Actually, you simply do the division as normal. 3x^3/2x = 3(x^2)/2. See? No issue here. The coefficients might get sort of messy, but division works normally.
• Is it possible to use synthetic division on a problem like this?
• No, it's not possible, because the divisor has an exponent higher than 1.
In order to divide polynomials using synthetic division, the denominator (the number(s) on the bottom of the fraction) must satisfy two rules:
1 - Be a linear expression, in other words, each term must either be a constant or the product of a constant and a single variable to the power of 1.
2 - The leading coefficient (first number) must be a 1.
For example, you can use synthetic division to divide a polynomial by (x + 2) or (x – 6), but you cannot use synthetic division to divide by 6x, or (2x + 3) or (3x^2 – x + 3).
• Is synthetic division the same as polynomial long division?
• i have a question that I am struggling with it asks... Using polynominal long division to determine the quotient when 3x^3 - 5x^2 + 10x + 4 is divided by 3x + 1

Any help with this would be greatly appreciated!
Thank you.
• well since the x values are already in descending order, you can start and you always use the first term in the divisor to determine what to multiply it by to get the dividend, so to get from 3x to 3x^3 you multiply by x^2 and then you have both terms of the divisor multiplied by that to get 3x^3 +x^2 which you subtract from 3x^3-5x^2 and you get -6x^2 and you bring down the 10x and to get from 3x to -6x^2 you multiply by -2x and you do that to both terms and you get -6x^2-2x which you subtract from -6x^2+10x to get 12x and you bring down the +4 and to get from 3x to 12x you multiply by 4 and 3x+1 times 4 is 12x+4 so when you subtract you get 0 and the answer is x^2-2x+4 (remainder 0)
• What would you do if the x term for the divisor is bigger than the dividend? Help would be much appreciated
• Then you have a proper fraction. It's like have 3/4. All you can do is factor the numerator and denominator to see if you can reduce the fraction.
• at , Sal canceled the denominator and is only left with the numerator. Why he doesn't distribute the x^2-x+1 into 5x-5 as well? Thanks!
• The entire fraction is one term, so he is distributing x^2-x+1 into the entire fraction, not just the denominator.
Let's say 5x-5 = a and x^2-x+1 = b. Distributing into the last term would leave you with (a/b) * b. Simplifying this would just leave you with a, or 5x-5.
This question was asked last year, so my answer is a bit late - you may have already figured it out yourself - but maybe it'll help with someone else who has this question, since there are no other answers here.