Question 174647
<font size = 7 color = "red"><b>Edwin's solution:</pre></b></font>
{{{(t^3-19t +9)/(t-4)}}} 
answer: {{{ matrix(1,3,t^2+4t-3, "",r=-3) }}}
<pre><font size = 4 color = "indigo"><b>

By inserting the term 0t<sup>2</sup> and using long division
or synthetic division.

Do you understand long division or synthetic division? 
If not post again asking how we get the following:

              t<sup>2</sup> +  4t -  3 
      --------------------- 
t - 4 ) t<sup>3</sup> - 0t<sup>2</sup> - 19t +  9
        t<sup>3</sup> - 4t<sup>2</sup>
        --------
             4t<sup>2</sup> - 19t
             4t<sup>2</sup> - 16t
             ---------
                    3t +  9
                    3t + 12 
                   --------
                         -3

Answer: t<sup>2</sup> + 4t - 3  r=-3

Or by synthetic division:

4 | 1  0  -19   9
  |    4   16 -12  
  ---------------
    1  4   -3  -3   

Answer: t<sup>2</sup> + 4t - 3  r=-3

Or if you know to put the remainder over the dividend:

Answer: {{{t^2 + 4t - 3 + (-3)/(t-4)}}} or {{{t^2 + 4t - 3 - 3/(t-4)}}} 

Edwin</pre>