Question 172908
I assume that you want to do long division right?



{{{(3x^4-2x^3+5x^2+x+1)/(x^2+2x)}}} Start with the given expression



Rewrite the expression like this:


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/1.png" alt="Photobucket - Video and Image Hosting">



Now the question is: How many times does {{{x^2}}} go into {{{3x^4}}}?



It goes in {{{3x^2}}} times. So place {{{3x^2}}} over the term {{{5x^2}}}.



<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/2.png" alt="Photobucket - Video and Image Hosting">



Now multiply {{{3x^2}}} by {{{x^2+2x}}} to get {{{3x^4+6x^3}}}



Reverse the signs of {{{3x^4+6x^3}}} to get {{{-3x^4-6x^3}}} (place this underneath the dividend)






<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/3.png" alt="Photobucket - Video and Image Hosting">




Now Add to get:


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/4.png" alt="Photobucket - Video and Image Hosting">




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Now the question is now: How many times does {{{x^2}}} go into {{{-8x^3}}}?



It goes in {{{-8x}}} times. Place this term over {{{x}}} in the dividend.



<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/5.png" alt="Photobucket - Video and Image Hosting">



Now multiply -8x by {{{x^2+2x}}} to get {{{-8x^3-16x^2}}}



Reverse the signs of {{{-8x^3-16x^2}}} to get {{{8x^3+16x^2}}} (place this underneath the last polynomial)



<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/6.png" alt="Photobucket - Video and Image Hosting">



Now add to get:


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/7.png" alt="Photobucket - Video and Image Hosting">



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Now how many times does {{{x^2}}} go into {{{21x^2}}}?



It goes in {{{21}}} times. Place this over the term {{{1}}} in the dividend.



<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/8.png" alt="Photobucket - Video and Image Hosting">



Now multiply 21 by {{{x^2+2x}}} to get {{{21x^2+42x}}}



Reverse the signs of {{{21x^2+42x}}} to get {{{-21x^2-42x}}}



<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/9.png" alt="Photobucket - Video and Image Hosting">



Now add to get:


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/10-2.png" alt="Photobucket - Video and Image Hosting">



Since {{{x^2}}} does NOT go into {{{-41x+1}}}. This means that the long division process stops.



So the quotient is {{{3x^2-8x+21}}} and the remainder is {{{-41x+1}}}




This means that {{{(3x^4-2x^3+5x^2+x+1)/(x^2+2x)=3x^2-8x+21+(-41x+1)/(x^2+2x)}}}