SOLUTION: how would you find "p" in the polynomial function: 8x^3+4x^2-"p"x+6 ...is divided by 2x-1 ....the remainder is 3 ....determine the value of p!!

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: how would you find "p" in the polynomial function: 8x^3+4x^2-"p"x+6 ...is divided by 2x-1 ....the remainder is 3 ....determine the value of p!!      Log On


   

Question 251543: how would you find "p" in the polynomial function:
8x^3+4x^2-"p"x+6 ...is divided by 2x-1 ....the remainder is 3 ....determine the value of p!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First write the problem out in long division form 
         _________________
    2x-1 | 8x^3+4x^2-px+6



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 


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


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

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



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. 

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



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 

         _______4x^2+4x__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6



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


         _______4x^2+4x__________
    2x-1 | 8x^3+4x^2-px+6
           8x^3-4x^2
           ---------
                8x^2-px+6
                8x^2-4x
                ---------



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

         _______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


Combine like terms. 

         _______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



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 



         _______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



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



         _______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)
                    --------------



Subtract the terms and distribute. 



         _______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


Combine like terms. 


         _______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



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%5E3%2B4x%5E2-10x%2B6 by 2x-1 will give you some quotient with a remainder of 3.