how do you find remainder from the details provided above?Quote:

If P(x) has the remainder -5 when divided by x-3 and the remainder 3 when divided by x+2, find the reminder when P(x) is divided by (x-3)(x+2)

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- Jan 14th 2009, 12:10 AMhungrybartsfinding P(x) and the remainderQuote:

If P(x) has the remainder -5 when divided by x-3 and the remainder 3 when divided by x+2, find the reminder when P(x) is divided by (x-3)(x+2)

- Jan 14th 2009, 12:27 AMMoo
Imagine you write the euclidean division :

There exists a polynomial Q such that :

$\displaystyle P(x)=(x-3)Q(x)-5$

Now let $\displaystyle x=3$

This gives $\displaystyle P(3)=0-5=-5$

Same thing for the division by (x+2) : we get $\displaystyle P(-2)=3$

Now what if we consider the division by $\displaystyle (x-3)(x+2)$ ?

There exists polynomials R and S (S is the remainder) such that :

$\displaystyle P(X)=(x-3)(x+2)R(x)+S(x)$

Now once again, if you let $\displaystyle x=3$, you get :

$\displaystyle \underbrace{P(3)}_{-5}=0+S(3)$ so we have $\displaystyle S(3)=-5$

Similarly, if you let $\displaystyle x=-2$, you get : $\displaystyle S(-2)=3$

So the remainder of P when divided by (x-2)(x+3) will be a polynomial S such that $\displaystyle S(3)=-5$ and $\displaystyle S(-2)=3$

But the degree of S cannot exceed 2, since (x-2)(x+3) has a degree 2. (if S has a degree equal or superior to 2, then it still can be divided by (x-2)(x+3))

Hence we're looking for a and b in :

$\displaystyle S(x)=ax+b$

Now the previous working gives you the following system :

$\displaystyle \left\{\begin{array}{ll} 3a+b=-5 \\ -2a+b=3 \end{array} \right.$

Which is very basic algebra :) - Jan 14th 2009, 12:30 AMhungrybarts
how if P(x) was a cubic equation? is it still possible to find P(x)

- Jan 14th 2009, 12:32 AMMoo
- Jan 14th 2009, 02:02 AMmr fantastic
Thread of related interest: http://www.mathhelpforum.com/math-he...mial-help.html

By the way, the question is asking for the remainder so I don't know why you keep talking about P(x). P(x) is**not**the remainder.