Long Division of PolynomialsDate: 26 Jun 1995 11:04:20 -0400 From: beth parrill Subject: Long division How do you use long division to solve polynomials with remainders? I am trying to help my mom figure it out for a boy that she tutors. He is in the 10th grade. Thanks... Beth Date: 26 Jun 1995 12:19:51 -0400 From: Dr. Ken Subject: Re: Long division Hello there! Here's an example of when you'd use polynomial long division. The steps are quite similar to the steps you do when dividing numbers, but that may not be helpful until you already know how to do it. Let's say we have the polynomial x^5 - 2x^2 + 4, and we want to divide it by x-4. Set it up like this: __________________________________ x-4 )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4 Then ask yourself "how many times does x (the highest power in the divisor) go into x^5 (the highest power in the dividend)?" In other words, what do you have to multiply x by to get x^5? Why, x^4, naturally. So do the multiplication (of the whole divisor) and write the factor you multiplied by on top: _______x^4________________________ x-4 )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4 x^5 - 4x^4 Now subtract from the dividend what you've just written down. To do that, I'll distribute a negative sign through what I've just written and add them: _______x^4________________________ x-4 )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4 -x^5 + 4x^4 ---------- 4x^4 Now bring down the next term in the dividend, and repeat the process: _______x^4_+_4x^3_________________ x-4 )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4 -x^5 + 4x^4 ---------- 4x^4 + 0x^3 4x^4 - 16x^3 ------------ (--subtract) 16x^3 Do it again: _______x^4_+_4x^3_+16x^2__________ x-4 )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4 -x^5 + 4x^4 ---------- 4x^4 + 0x^3 4x^4 - 16x^3 ------------ 16x^3 + 2x^2 16x^3 - 64x^2 ------------- (--subtract) 66x^2 Again: _______x^4_+_4x^3_+16x^2_+66x_____ x-4 )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4 -x^5 + 4x^4 ---------- 4x^4 + 0x^3 4x^4 - 16x^3 ------------ 16x^3 + 2x^2 16x^3 - 64x^2 ------------- 66x^2 + 0x 66x^2 -264x ----------- 264x One more time: _______x^4_+_4x^3_+16x^2_+66x_+264___ x-4 )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4 -x^5 + 4x^4 ---------- 4x^4 + 0x^3 4x^4 - 16x^3 ------------ 16x^3 + 2x^2 16x^3 - 64x^2 ------------- 66x^2 + 0x 66x^2 -264x ----------- 264x + 4 264x - 1056 ----------- 1060 Yay! So the answer is that x^5 + 2x^2 + 4 divided by x-4 is x^4 + 4x^3 + 16x^2 + 66x + 264, with a remainder of 1060. Hope that helps, and if you have any questions about this, please ask us. -K |
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