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!!

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) (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

So dividing the polynomial by will give you some quotient with a remainder of 3.

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