# find values

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• Sep 19th 2009, 05:42 AM
mark
find values
hi, i've got the question:

the polynomials f(x) and g(x) are defined by $\displaystyle f(x) = x^3 + px^2 - x + 5$, $\displaystyle g(x) = x^3 - x^2 + px + 1$ where p is a constant. when f(x) and g(x) are divided by x - 2 the remainder is R in each case. find the values of p and R

i got it down to
R = f(2) = 8 + 4p - 2 + 5 = 11 + 4p and
R = g(2) = 8 - 4 + 2p + 1 = 5 + 2p but i'm not sure if either of those are right.

can someone show me the way to do it please? thanks
• Sep 19th 2009, 05:48 AM
VonNemo19
Quote:

Originally Posted by mark
hi, i've got the question:

the polynomials f(x) and g(x) are defined by $\displaystyle f(x) = x^3 + px^2 - x + 5$, $\displaystyle g(x) = x^3 - x^2 + px + 1$ where p is a constant. when f(x) and g(x) are divided by x - 2 the remainder is R in each case. find the values of p and R

i got it down to
R = f(2) = 8 + 4p - 2 + 5 = 11 + 4p and
R = g(2) = 8 - 4 + 2p + 1 = 5 + 2p but i'm not sure if either of those are right.

can someone show me the way to do it please? thanks

I would start by dividing each by x-2, then set the remainders equal.
• Sep 19th 2009, 05:52 AM
galactus
$\displaystyle \frac{x^{3}+px^{2}-x+5}{x-2}=\underbrace{\frac{4p+11}{x-2}}_{\text{remainder}}$

$\displaystyle \frac{x^{3}-x^{2}+px+1}{x-2}=\underbrace{\frac{2p+5}{x-2}}_{\text{remainder}}$

$\displaystyle 4p+11=2p+5\Rightarrow p=-3$

$\displaystyle R=-1$

Therefore, the remainder in each case would be $\displaystyle \frac{-1}{x-2}$

$\displaystyle \frac{x^{3}-x^{2}-3x+1}{x-2}=\boxed{\frac{-1}{x-2}}+x^{2}+x-1$

$\displaystyle \frac{x^{3}-3x^{2}-x+5}{x-2}=\boxed{\frac{-1}{x-2}}+x^{2}-x-3$
• Sep 19th 2009, 06:48 AM
pacman
Galactus, your latexing method is as good as that of soroban, excellent!
• Sep 19th 2009, 07:45 AM
mark
ok i understand it now up to $\displaystyle \frac {-1}{x - 2}$, when you get to that, wouldn't x have to be 3 for the final answer to be -1? how do you know the value of x?
• Sep 19th 2009, 08:33 AM
galactus
Yes, p=-3. Sub that in and we get R=-1