MEDAL AND FAN! Please help, quick problem!

Question

anonymous · Answer

attachment

anonymous · Answer

first say what is x^2 - 16? can u simplify it?

anonymous · Answer

(x+4)(-4) ?

aum · Answer

(x+4)(x-4). Factor (x^2 + 5x + 6) too.

anonymous · Answer

(x+3)(x+2)

aum · Answer

Cancel common factors.

anonymous · Answer

correct ;) now plug in that factors in that expression and see what will happen.

anonymous · Answer

(x+4)/(x+3)(x+2)*(x+3)/(x+4)(x-4) 
So the only thing left would be (x+2)*(x+4)  ? after common factors

anonymous · Answer

yes but x-4 :

$$\frac{ 1 }{ (x+2)(x-4) }$$

anonymous · Answer

Oh, right, my handwriting messed me up. So that is obviously the end, right? lol

anonymous · Answer

yes ;)

anonymous · Answer

And it also says to prove closure?

anonymous · Answer

to prove what?

anonymous · Answer

Closure

aum · Answer

The result you got dividing one polynomial by another is not a polynomial.
So polynomial division is not closed.

anonymous · Answer

yes i know but we don't use "prove" for closure and @aum explained what is closure.

aum · Answer

If you add, subtract, multiply two polynomials the result will also be a polynomial.  Therefore polynomials are closed under addition, subtraction and multiplication.

That is not true, in general, for division of polynomials.

anonymous · Answer

Thank you! @aum @PFEH.1999