How do you divide #(x^3+x^2+3x+1)/(x-3)#?

2 Answers
Jul 14, 2017

#x^2 +4x+15 " rem " 46/(x-3)#

Explanation:

Division by synthetic division is the simplest and quickest method in this case. We will only use the numerical coefficients.

#x-3 =0 " "rarr x =3" "# goes outside on the left.

#(1x^3 +1x^2 +3x+1)/(x-3)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#" "3|" "1" "1" "3" "1#
#" "|ul" "darrul" "#
#" "1color(white)(xxxxxxxxxxxxxxx)larr# bring down the #1#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#" "color(red)(3)|" "1" "1" "3" "1#
#" "|ul" "darrul" "color(blue)(3)ulcolor(white)(xxxxxxxxxxx)larrcolor(red)(3xx1) = color(blue)(3)#
#" "color(red)(1)" "color(blue)(4)color(white)(xxxxxxxxxxx)larr 1+color(blue)(3 =4)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#" "color(red)(3)|" "1" "1" "3" "1#
#" "|ul" "darrul" "3ul" "color(green)(12)ulcolor(white)(xxxxxxxxx)larrcolor(red)(3xxcolor(blue)(4) color(green)(=12)#
#" "1" "color(blue)(4)" "color(green)(15)color(white)(xxxxxxxxx)larr 3+color(green)(12 =15)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#" "color(red)(3)|" "1" "1" "3" "1#
#" "|ul" "darrul" "3ul" "12" "color(magenta)(45)ulcolor(white)(xxxxxx)larrcolor(red)(3xxcolor(green)(15) color(magenta)(=45)#
#" "1" "color(blue)(4)" "color(green)(15)" "color(magenta)(46)color(white)(xxxxxx)larr 1+color(magenta)(45 =46)#
#color(white)(xxxxxxxxxxxxxxxx)uarr#
#color(white)(xxxxxxxxxxxxxxx)"remainder"#

The bottom line gives the numerical coefficients of the quotient.
#x^3 divx =x^2#

Quotient:

#1x^2 +color(blue)(4)x +color(green)15" rem " color(magenta)(46)#

Jul 14, 2017

The remainder is #46# and the quotient is #=x^2+4x+15#

Explanation:

Let's perform the synthetic division

#color(white)(aaaa)##3##color(white)(aaaa)##|##color(white)(aaaa)##1##color(white)(aaaa)##1##color(white)(aaaaaa)##3##color(white)(aaaaaaa)##1#
#color(white)(aaaaaaaaaaaa)#_________

#color(white)(aaaa)##color(white)(aaaaaaa)##|##color(white)(aaaa)##color(white)(aaa)##3##color(white)(aaaaa)##12##color(white)(aaaaaa)##45#
#color(white)(aaaaaaaaaaaa)#________

#color(white)(aaaa)##color(white)(aaaaaaa)##|##color(white)(aaa)##1##color(white)(aaa)##4##color(white)(aaaaa)##15##color(white)(aaaaaa)##color(red)(46)#

The remainder is #46# and the quotient is #=x^2+4x+15#

#(x^3+3x^2+3x+1)/(x-3)=x^2+4x+15+46/(x-3)#