Rational Expressions- Simplifying

Hello fellow students. Some of you may have unfortunately not been able to attend today's class. Don't worry, I will tell you everything you will need to know for next class. Today we started off with stating the restriction on the variable. We learned that the denominator could never be zero. That is because when zero is the numerator it is divided and equal to the number. But does that mean that when a number is divided by zero it is what the number is? No, whenever the zero is the denominator, we say it is undefined.
(ex. 0/7= 0, 7/0= undefined)

x-4/x²-16 was one of the questions we had to state the restriction. When we simplify it, it becomes x-4/ (x-4)(x+4). We then reduce x-4 from both the numerator and the denominator. We are now left with 1/x+4. x cannot equal to 4 or -4. The reason why x cannot equal 4 or -4 is because when youfactorr them in, the denominator will equal to zero. It's okay to factor 4 in the result, that's because it won't be undefined, it'll be 1/8.
You don't have the right answer unless you state the restriction of the variable.

x-4/x²-16
=x-4/ (x-4)(x+4)
=1/x+4
x cannot equal 4 or -4

The class then moved on to simplifying rational equations 2x²+x-15/ 3x+9 was one of the questions we had simplify. When you simplify the equation you should get (2x-5)(x+3)/ 3(x+3). x+3 reduces and your left with 2x-5/ 3. x cannot equal -3, that's because when you factor it you would get 2x²+x-15/ 3(-3)+9. The denominator would be zero.

2x²+x-15/ 3x+9
=(2x-5)(x+3)/ 3(x+3)
=2x-5/ 3
x cannot equal -3

You could also factor out of a negative. It will make a little change, it will change the signs.
x²-1/ 1-x
=(x+1)(x-1)/(1-x)
=(x+1)(x-1)/-(-1+x)
=(x+1)(x-1)/-(x-1)
=x+1/-1
=-x - 1
x cannot equal 1

Mr. Kuropatwa had also refreshed us on factoring. He showed us a neat algerbraic trick to factor instead of guess and check. First you multiply the first number with the last. We then factor out the common factor by the groups we made. We then are left with the same numbers in the braket. We then take the common factors from outside and put them together.

2x²+x-15
[(2)(-15) = -30]
[-30/6=-5]
=2x²+6x-5x-15
=2x(x+3) -5(x+3)
=(x+3)(2x-5)
it's in your dictionary
*NEVER CANCEL ANYTHING, WE JUST REDUCE! A rule Mr. K is passionate about.

For homework we do excercise 43 omitting numbers 11 and 12. Well, I hope I explained this clearly for those who were lost or did not attend the class. If you have any questions, ask your fellow classmates or you teacher, Mr. Kuropatwa. I know he'll be happy that you asked. That's about it for me, the scribe for tommorrow is Thang_N. Good luck at being scribe tommorrow!

Yours Truely
SAMUS