Algebra (all content)
Dividing polynomials: long division
Sal divides (x^2-3x+2) by (x-2) and then checks the solution. Created by Sal Khan and Monterey Institute for Technology and Education.
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- Is there a video on synthetic division.(22 votes)
- Sal has made some videos about it now:
- Hi, I'm stuck on this question from school...
When P(x) = x^4 + ax^2 + bx + 2 is divided by x^2 +1 the remainder is -x+1. Find the values of a and b.
Thank you!(7 votes)
- If x⁴ + ax² + bx + 2 divided by (x²+1) leaves a remainder of -x + 1, then:
(x²+1) divides x⁴ + ax² + bx + 2 - (-x +1) = x⁴ + ax² + (b + 1) + 1 exactly.
Since (x²+1) = (x + i)(x - i) this tells us (x - i) also divides x⁴ + ax² + (b + 1)x + 1 and, by the Polynomial Remainder Theorem, i is a zero.
Substituting x = i in to x⁴ + ax² + (b + 1)x + 1 = 0 gives:
1 - a + (b + 1)i + 1 = 0
And by comparing real and imaginary parts we get a = 2, and b = -1(19 votes)
- How do you divide a monomial by a polynomial? Thank you!!(5 votes)
- With the monomial in the numerator, your only option is to factor the denominator and cancel out any common factors shared with the numerator.(14 votes)
- What do you call a 9 side polygon?(5 votes)
- As Ellie said, it's a nonagon.
Alternatively, especially for polygons with a LOT of sides, you can just call them a n-gon, with n being the number of sides. So a less common way to call a nonagon is a 9-gon, while a 129 sided polygon would just normally be called a 129-gon.(4 votes)
- Could you use (x^2/x-2) - (3x/x-2) + (2/x-2). I can't seem to find a way to divide a monomial over a polynomial. Is it possible?(5 votes)
- Well, they have the same denominator, so you have (x^2 -3x +2)/(x-2). Then just model what they do in this video.
OR, you can notice that (x^2 -3x +2) = (x-2)(x-1), so:
(x^2 -3x +2)/(x-2) = (x-2)(x-1)/(x-2) =x-1 (where x != 2)(3 votes)
- What if you have a problem where the denominator has variables raised to higher exponents than the numerator has? I see a lot of those in calculus.(5 votes)
- Then there's no such possible division. There's no integer number that multiplied by the denominator would result in the numerator. Depends on what part of calculus you're studying, we would have to see how the function behaves and apply limits or other concept proper from calculus. But is not so related to this.(1 vote)
- I kinda feel i am the only one here lol, all the comments are from 5 to 10 years!(3 votes)
- You are seeing the top ranked questions and answers. This is the default sort order. To see the newer ones, select the "recent" option to see the more recent questions and answers.(3 votes)
- At1:13it was -3x minus 2x, which would be -5x. But when he multiplied by -1 it turned into -3 plus 2x. Is that a mistake or does it work somehow? This confuses me...(2 votes)
- I love the Keepers of the Lost Cities too! (Sorry for how off topic this is...)(5 votes)
- this stuff kills my brain i'm just sayin'!
(Polynomial long division)
(y 3 - y 2 + y + 3) ÷ (y + 1)
GOD PLEASE HELP MEEEEEE!(4 votes)
- Can you show how to divide polynomials when all terms has an exponent? =((4 votes)
- that you should ask directly to Sal himself or one of the workers of this site(0 votes)
Divide x squared minus 3x plus 2 divided by x minus 2. So we're going to divide this into that. And we can do this really the same way that you first learned long division. So we have x minus 2 being divided into x squared minus 3x plus 2. Another way we could have written the same exact expression is x squared minus 3x plus 2, all of that over x minus 2. That, that, and that are all equivalent expressions. Now, to do this type of long division-- we can call it algebraic long division-- you want to look at the highest degree term on the x minus 2 and the highest degree term on the x squared minus 3x plus 2. And here's the x, and here's the x squared. x goes into x squared how many times? Or x squared divided by x is what? Well, that's just equal to x. So x goes into x squared x times. And I'm going to write it in this column right here above all of the x terms. And then we want to multiply x times x minus 2. That gives us-- x times x is x squared. x times negative 2 is negative 2x. And just like you first learned in long division, you want to subtract this from that. But that's completely the same as adding the opposite, or multiplying each of these terms by negative 1 and then adding. So let's multiply that times negative 1. And negative 2x times negative 1 is positive 2x. And now let's add. x squared minus x squared-- those cancel out. Negative 3x plus 2x-- that is negative x. And then we can bring down this 2 over here. So it's negative x plus 2 left over, when we only go x times. So then we say, can x minus 2 go into negative x plus 2? Well, x goes into negative x negative one times. You can look at it right here. Negative x divided by x is negative 1. These guys cancel out. Those guys cancel out. So negative 1 times x minus 2-- you have negative 1 times x, which is negative x. Negative 1 times negative 2 is positive 2. And we want to subtract this from that, just like you do in long division. But that's the same thing as adding the opposite, or multiplying each of these terms by negative 1 and then adding. So negative x times negative 1 is positive x. Positive 2 times negative 1 is negative 2. These guys cancel out, add up to 0. These guys add up to 0. We have no remainder. So we got this as being equal to x minus 1. And we can verify it. If we multiply x minus 1 times x minus 2, we should get this. So let's actually do that. So let's multiply x minus 1 times x minus 2. So let's multiply negative 2 times negative 1. That gives us positive 2. Negative 2 times x-- that's negative 2x. Let's multiply x times negative 1. That is negative x. And then x times x is x squared. And then add all the like terms. x squared, negative 2x minus x-- that's negative 3x. And then 2 plus nothing is just 2. And so we got that polynomial again.