SOLUTION: please help Factor completely Remember to look first for a common factor. Check by multiplying. If a polynomial is prime, state this x^2 - 2/5 x + 1/25

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: please help Factor completely Remember to look first for a common factor. Check by multiplying. If a polynomial is prime, state this x^2 - 2/5 x + 1/25       Log On


   

Question 623134: please help
Factor completely Remember to look first for a common factor. Check by multiplying. If a polynomial is prime, state this
x^2 - 2/5 x + 1/25

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2+-+%282%2F5%29x+%2B+1%2F25
The greatest common factor here is 1. And we rarely factor 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 patters 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. The a%5E2%2B2ab%2Bb%5E2 has two additions. Yours has a subtraction in the middle so a%5E2%2B2ab%2Bb%5E2 cannot be used either.

But the last one, a%5E2-2ab%2Bb%5E2 has the right pattern of addition and subtraction so it might work. Next we check to see if each individual term of your expression also matches each term of the pattern.

The first term of of the pattern is a%5E2. Is the first term of your expression a perfect square? Answer: Yes. It is x%5E2. This makes your "a" an "x".

The third term of of the pattern is b%5E2. Is the third term of your expression a perfect square? Answer: Yes. It is %281%2F5%29%5E2. This makes your "b" a "1/5".

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%28x%29%2A%281%2F5%29+=+%282%2F5%29x
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
x%5E2+-+%282%2F5%29x+%2B+1%2F25
into
%28x-1%2F5%29%5E2