How do you divide #(x^5 + x^3 − 9)/(x-1) #?
1 Answer
#(x^5+x^3-9)/(x-1) = (x^4+x^3+2x^2+2x+2)-7/(x-1)#
Explanation:
You can long divide the coefficients, not forgetting to include
In our example we find:
#x^5+x^3-9 = (x-1)(x^4+x^3+2x^2+2x+2)-7#
or if you prefer:
#(x^5+x^3-9)/(x-1) = (x^4+x^3+2x^2+2x+2)-7/(x-1)#
That is:
If all we wanted to know was the remainder, then we could have substituted
Alternative Method
Alternatively, start writing out the answer linearly, evaluating the partial sums as you go along and choosing the next term accordingly:
#x^5+x^3-9#
#(x-1)(x^4 color(white)(xxxxxxxxxxxxxxx)# gives#x^5-x^4#
#(x-1)(x^4+x^3 color(white)(xxxxxxxxxxx)# gives#x^5-x^3#
#(x-1)(x^4+x^3+2x^2 color(white)(xxxxxxx)# gives#x^5+x^3-2x^2#
#(x-1)(x^4+x^3+2x^2+2x color(white)(xxxx)# gives#x^5+x^3-2x#
#(x-1)(x^4+x^3+2x^2+2x+2) color(white)(x)# gives#x^5+x^3-2#
#(x-1)(x^4+x^3+2x^2+2x+2) -7 color(white)(x)# gives#x^5+x^3-9#