SOLUTION: please help I dont understand this at all theres two problems I need help with here is what it says Factor completely Remember to look first for a common factor. Check by multip

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: please help I dont understand this at all theres two problems I need help with here is what it says Factor completely Remember to look first for a common factor. Check by multip      Log On


   

Question 623133: please help I dont understand this at all theres two problems I need help with here is what it says
Factor completely Remember to look first for a common factor. Check by multiplying. If a polynomial is prime, state this
-81p^2 + 18pq - q^2


Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
-81p%5E2+%2B+18pq+-+q%5E2
The greatest common factor here is 1. And we rarely bother factoring out a 1.

Now we move on to other factoring techniques. One of these techniques is to use a factoring patterm:
  • Difference of squares: a%5E2-b%5E2+=+%28a%2Bb%29%28a-b%29
  • Sum of cubes: a%5E3%2Bb%5E3+=+%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29
  • Difference of cubes: a%5E3-b%5E3+=+%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29
  • Perfect square trinomials:
    • a%5E2%2B2ab%2Bb%5E2=+%28a%2Bb%29%5E2 [or (a+b)(a+b)]
    • a%5E2-2ab%2Bb%5E2=+%28a-b%29%5E2 [or (a-b)(a-b)]
The first three patterns have two terms on the un-factored side. Your expression has three terms so we cannot use them.

The last two patterns have three terms so they may work. Your expression has a pattern of "-" then "+" then "-". Neither of the last two patterns have this pattern of pluses and minuses. But... your pattern of pluses and minuses is the opposite of the pattern of pluses and minuses in the a%5E2-2ab%2Bb%5E2 pattern. So by factoring out -1, we can match the pluses and minuses of a%5E2-2ab%2Bb%5E2:
%28-1%29%2881p%5E2+-+18pq+%2B+q%5E2%29

Next we check to see if each individual term of 81p%5E2+-+18pq+%2B+q%5E2 also matches each term of the a%5E2-2ab%2Bb%5E2 pattern.

The first term of of the pattern is a%5E2. Is your first term, 81p%5E2, a perfect square? Answer: Yes. It is %289p%29%5E2. This makes your "a" a "9p".

The third term of of the pattern is b%5E2. Is your third term, q%5E2 a perfect square? Answer: Obviously. This makes your "b" a "q".

And finally, the second term of the pattern is 2ab. Is the second term of your expression equal to 2 times your "a" times your "b"? Since your "a" is x and your "b" is 1/5, 2 times your "a" times your "b" becomes:
2%2A%289p%29%2A%28q%29+=+18pq
So, yes, your second term also matches the pattern. So we have a match!! (Note: If the additions and subtractions did not match or if any of the terms had not matched the pattern then we would have stopped trying to use this pattern. And since this was the last pattern, we would then have to move on to other factoring techniques.)

Now we can use the pattern (with an "a" of "x" and a "b" of "1/5") to factor
81p%5E2+-+18pq+%2B+q%5E2
into
%289p-q%29%5E2

So
-81p%5E2+%2B+18pq+-+q%5E2
factors into:
%28-1%29%289p-q%29%5E2

Note: When you're factoring and you have a pattern of pluses and minuses that are not working, keep in mind that you can always factor out a -1 (which has the effect of flipping all the pluses to minuses and vice versa. We could have done this problem without factoring out the -1. But this way was easier so it is good to keep this in mind.