Question 251543
First write the problem out in long division form

<pre>
         _________________
    2x-1 | 8x^3+4x^2-px+6

</pre>



Now ask yourself: what must you multiply 2x by to get 8x^3? The answer is 4x^2. Place this above the x^2 term



<pre>
         _______4x^2__________
    2x-1 | 8x^3+4x^2-px+6

</pre>


Multiply 4x^2 by 2x-1 to get 8x^2-4x^2 and place that beneath the first two terms


<pre>
         _______4x^2__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
</pre>



Subtract the bottom expression 8x^3-4x^2 from the top expression 8x^3+4x^2 to get 8x^2 and bring down the rest of the terms.


<pre>
         _______4x^2__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
</pre>



This is where we start back from the beginning and ask: what expression can you multiply 2x by to get 8x^2? The answer is 4x. Place this above the -px term


<pre>
         _______4x^2+4x__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
</pre>



Multiply 4x by 2x-1 to get 8x^2-4x. Place this underneath 8x^2-px+6



<pre>
         _______4x^2+4x__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
                8x^2-4x
                ---------
</pre>



Subtract 8x^2-4x from 8x^2-px+6


<pre>
         _______4x^2+4x__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
                8x^2-4x
                ---------
                    -px+4x+6
</pre>


Combine like terms.


<pre>
         _______4x^2+4x__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
                8x^2-4x
                ---------
                    (-p+4)x+6
</pre>



Now onto the final stage: what expression must you multiply 2x by to get (-p+4)x? The answer is (1/2)(-p+4). Write this above the last term 6




<pre>
         _______4x^2+4x+(1/2)(-p+4)__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
                8x^2-4x
                ---------
                    (-p+4)x+6
</pre>



Multiply (1/2)(-p+4) by 2x-1 to get (-p+4)x-(1/2)(-p+4)




<pre>
         _______4x^2+4x+(1/2)(-p+4)__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
                8x^2-4x
                ---------
                    (-p+4)x+6
                    (-p+4)x-(1/2)(-p+4)
                    --------------
</pre>



Subtract the terms and distribute.




<pre>
         _______4x^2+4x+(1/2)(-p+4)__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
                8x^2-4x
                ---------
                    (-p+4)x+6
                    (-p+4)x-(1/2)(-p+4)
                    --------------
                          6-(1/2)p+2
</pre>


Combine like terms.



<pre>
         _______4x^2+4x+(1/2)(-p+4)__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
                8x^2-4x
                ---------
                    (-p+4)x+6
                    (-p+4)x-(-p+4)
                    --------------
                          8-(1/2)p
</pre>



So the remainder is 8-(1/2)p. But we're given that the remainder is 3. So this means that 8-(1/2)p=3 and its solution is p=10.



So the value of p is {{{p=10}}}



So dividing the polynomial {{{8x^3+4x^2-10x+6}}} by {{{2x-1}}} will give you some quotient with a remainder of 3.