SOLUTION: Hi, good afternoon. Can you please help me with these two problems. I do not seem to understand how to: Divide and simplify: {{{(3x^2+13x+4)/(16-x^2)}}}÷{{{(3x^2-5x-12)/(3x-12

Algebra ->  Radicals -> SOLUTION: Hi, good afternoon. Can you please help me with these two problems. I do not seem to understand how to: Divide and simplify: {{{(3x^2+13x+4)/(16-x^2)}}}÷{{{(3x^2-5x-12)/(3x-12      Log On


   

Question 89827: Hi, good afternoon.
Can you please help me with these two problems.
I do not seem to understand how to:
Divide and simplify: %283x%5E2%2B13x%2B4%29%2F%2816-x%5E2%29÷%283x%5E2-5x-12%29%2F%283x-12%29
Simplify by rationalizing the denominator:
%28sqrt%2811%29-sqrt%285%29%29+%2F+%28sqrt%2811%29%2Bsqrt%285%29%29
Thank you so much,
Chocolatechrrys
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

Hi, good afternoon. 
Can you please help me with these two problems.
I do not seem to understand how to: 
Divide and simplify: %283x%5E2%2B13x%2B4%29%2F%2816-x%5E2%29÷%283x%5E2-5x-12%29%2F%283x-12%29 
Simplify by rationalizing the denominator: 
%28sqrt%2811%29-sqrt%285%29%29+%2F+%28sqrt%2811%29%2Bsqrt%285%29%29
Thank you so much,
Chocolatechrrys 

============================================

%283x%5E2%2B13x%2B4%29%2F%2816-x%5E2%29÷%283x%5E2-5x-12%29%2F%283x-12%29 

Let's get each polynomial in order of descending powers of x.
Only one isn't, namely 16-x%5E2. So write the 16-x%5E2 
in order of descending powers of x,  that is, -x%5E2%2B16

Now factor -1 out of that -1%28x%5E2-16%29

%283x%5E2%2B13x%2B4%29%2F%28-1%28x%5E2-16%29%29÷%283x%5E2-5x-12%29%2F%283x-12%29 

We can erase the -1 and put a negative sign in front of the
first fraction

-%283x%5E2%2B13x%2B4%29%2F%28x%5E2-16%29÷%283x%5E2-5x-12%29%2F%283x-12%29 

Factor everything:

-%28%28x%2B4%29%283x%2B1%29%29%2F%28%28x-4%29%28x%2B4%29%29÷%28%283x%2B4%29%28x-3%29%29%2F%283%28x-4%29%29 


Invert the second fraction and change the ÷ to multiplication

-%28%28x%2B4%29%283x%2B1%29%29%2F%28%28x-4%29%28x%2B4%29%29·%283%28x-4%29%29%2F%28%283x%2B4%29%28x-3%29%29

Indicate the multiplication of the numerators and denominator
all as one fraction:

-%283%28x%2B4%29%283x%2B1%29%28x-4%29%29%2F%28%28x-4%29%28x%2B4%29%283x%2B4%29%28x-3%29%29

The %28x%2B4%29's will cancel, and also the %28x-4%29's

So we end up with

-%283%283x%2B1%29%29%2F%28%283x%2B4%29%28x-3%29%29

You can leave that just asit is, or if you like,
you can multiply the numerator and denominator out:

-%289x%2B3%29%2F%283x%5E2-5x-12%29

As you see, we could have gotten by without
factoring the 3x%5E2-5x-12 but we had no way
or knowing in advance that neither of its factors
would cancel, and they might have.

=========================================

%28sqrt%2811%29-sqrt%285%29%29+%2F+%28sqrt%2811%29%2Bsqrt%285%29%29

Put parentheses around the top and bottom:

%28%28sqrt%2811%29-sqrt%285%29%29%29+%2F+%28%28sqrt%2811%29%2Bsqrt%285%29%29%29

Form the conjugate of the denominator by
copying it over and changing the sign of
only the second term.  So the conjugate
of denominator sqrt%2811%29%2Bsqrt%285%29 is
sqrt%2811%29-sqrt%285%29. Now multiply by the 
fraction 

%28%28sqrt%2811%29-sqrt%285%29%29%29%2F%28%28sqrt%2811%29-sqrt%285%29%29%29, which just equals 1.

%28%28sqrt%2811%29-sqrt%285%29%29%29+%2F+%28%28sqrt%2811%29%2Bsqrt%285%29%29%29·%28%28sqrt%2811%29-sqrt%285%29%29%29%2F%28%28sqrt%2811%29-sqrt%285%29%29%29

Indicating the multiplication of numerators and 
denominators:



Now use FOIL to multiply out:



Multiplying under the radicals

%2811-sqrt%2855%29-sqrt%2855%29%2B5%29%2F%2811-sqrt%2855%29%2Bsqrt%2855%29-5%29

The two middle terms in the top combine as -2sqrt%2855%29,
and the two middle terms cancel in the bottom:

%2811-2sqrt%2855%29%2B5%29%2F%2811-5%29

Combining the numbers 

%2816-2sqrt%2855%29%29%2F6

Factor 2 out of the top:

%28+2%288-sqrt%2855%29%29+%29%2F6

Cancel the 2 into the 6 and get

+%288-sqrt%2855%29%29+%2F3

Edwin