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## Class 10 (Foundation)

### Course: Class 10 (Foundation)>Unit 3

Lesson 3: Division of polynomials

# 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.

## Want to join the conversation?

• Is there a video on synthetic division.
• 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!
• 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
• How do you divide a monomial by a polynomial? Thank you!!
• With the monomial in the numerator, your only option is to factor the denominator and cancel out any common factors shared with the numerator.
• What do you call a 9 side polygon?
• 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.
• 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?
• 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)
• 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.
• 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!
• 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.
• At it 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...
• I love the Keepers of the Lost Cities too! (Sorry for how off topic this is...)
• this stuff kills my brain i'm just sayin'!

(Polynomial long division)
(y 3 - y 2 + y + 3) ÷ (y + 1)