1. ## polynomials

For what value of b will the polynomial
P(x) = -2x^3+bx^2 -5x +2 have the same remainder when it is divided by x-2 and x + 1?

I thought of finding the remainder by using the remainder theorm and combine them both, and i thought of foiling it, but that wont work. u can't use divide it since b is still unkown. what do u do!! lol thanks

2. Originally Posted by lickman
For what value of b will the polynomial
P(x) = -2x^3+bx^2 -5x +2 have the same remainder when it is divided by x-2 and x + 1?

I thought of finding the remainder by using the remainder theorm and combine them both, and i thought of foiling it, but that wont work. u can't use divide it since b is still unkown. what do u do!! lol thanks
by the remainder theorem, we need P(2) = P(-1)

3. P(2) = P(-1) , so do u subsititue both of these to x? do u mean that both of these will result in the same remainder?

4. Originally Posted by lickman
P(2) = P(-1) , so do u subsititue both of these to x? do u mean that both of these will result in the same remainder?
the remainder theorem says, that if a polynomial P(x) is divided by (x - a) then the remainder is P(a)

so the remainder when P(x) is divided by x - 2 is P(2) and the remainder when P(x) is divided by x + 1 is P(-1). we want these to be equal, so we set them equal.

P(2) means you replace all the x's in P(x) with 2 and simplify. do the similar thing for P(-1). set the expressions equal to each other. you will have a linear equation in b, solve for b

5. Originally Posted by lickman
For what value of b will the polynomial
P(x) = -2x^3+bx^2 -5x +2 have the same remainder when it is divided by x-2 and x + 1?

I thought of finding the remainder by using the remainder theorm and combine them both, and i thought of foiling it, but that wont work. u can't use divide it since b is still unkown. what do u do!! lol thanks
You could use synthetic division:

The synthetic division process for $\displaystyle \frac{-2x^3+bx^2-5x+2}{x-2}$

Code:
__
2|   -2     b      -5         2
-4     2b-8      4b-26
-------------------------------
-2    b-4    2b-13    |4b-24
We have a remainder of $\displaystyle 4b-24$

Now, the synthetic division process for $\displaystyle \frac{-2x^3+bx^2-5x+2}{x+1}$

Code:
__
-1|   -2     b      -5         2
2     -b-2       b+7
-------------------------------
-2    b+2    -b-7      |b+9
We have a remainder of $\displaystyle b+9$

Now, we want a value of $\displaystyle b$ that gives us the same remainder:

$\displaystyle 4b-24=b+9\implies 3b=33\implies \color{red}\boxed{b=11}$

I hope this makes sense!

--Chris

6. lol hey, thanks for the effort really appreciate but i never learned the synthesis divison lol. so can u use the P(2) = P (-1) and show me haha thanks!?!?!?!

7. oh nvm, how stupid of me, i understand it thanks! i get it now LOL