How do you find the quotient of #(x^5-x^4+3x^3+x^2)-:(x^3+x^2+1)#?

1 Answer
Douglas K.
Jan 2, 2018

The quotient is #x^2-2x+5# with a remainder of #-5x^2+2x-5#

Please see the explanation for a description of how to perform the long division.

Explanation:

Starting with:
#color(white)( (x^3+x^2+1)/color(black)(x^3+x^2+1))color(white)((x^5-x^4+3x^3+x^2))/(")" color(white)(x)x^5-x^4+3x^3+x^2)#

Insert the terms with a 0 coefficient in both the divisor and dividend:

#color(white)( (x^3+x^2+0x+1)/color(black)(x^3+x^2+0x+1))color(white)((x^5-x^4+3x^3+x^2+0x+0))/(")" color(white)(x)x^5-x^4+3x^3+x^2+0x+0)#

The first term in the quotient is #x^2#, because, when multiplied by the divisor, it will allow us to eliminate the #x^5# term in the dividend:

#color(white)( (x^3+x^2+0x+1)/color(black)(x^3+x^2+0x+1))(x^2color(white)(x^4+3x^3+x^2+0x+0))/(")" color(white)(x)x^5-x^4+3x^3+x^2+0x+0)#
#color(white)(............................)ul(-x^5-x^4-0x^3-x^2)#
#color(white)(.................................)-2x^4+3x^3-0x^2+0x#

The next term in the quotient is #-2x#, because, when multiplied by the divisor, it will allow us to eliminate the #-2x^4# term in the dividend:

#color(white)( (x^3+x^2+0x+1)/color(black)(x^3+x^2+0x+1))(x^2-2xcolor(white)(3x^3+x^2+0x+0))/(")" color(white)(x)x^5-x^4+3x^3+x^2+0x+0)#
#color(white)(............................)ul(-x^5-x^4-0x^3-x^2)#
#color(white)(.................................)-2x^4+3x^3+0x^2+0x#
#color(white)(.....................................)ul(2x^4+2x^3+0x^2+2x)#
#color(white)(..............................................)5x^3+0x^2+2x+0#

The last term in the quotient is #5#, because, when multiplied by the divisor, it will allow us to eliminate the #5x^3# term in the dividend:

#color(white)( (x^3+x^2+0x+1)/color(black)(x^3+x^2+0x+1))(x^2-2x+5color(white)(+x^2+0x+0))/(")" color(white)(x)x^5-x^4+3x^3+x^2+0x+0)#
#color(white)(............................)ul(-x^5-x^4-0x^3-x^2)#
#color(white)(.................................)-2x^4+3x^3+0x^2+0x#
#color(white)(.....................................)ul(2x^4+2x^3+0x^2+2x)#
#color(white)(..............................................)5x^3+0x^2+2x+0#
#color(white)(...........................................)ul(-5x^3-5x^2-0x-5)#
#color(white)(....................................................)-5x^2+2x-5#

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