Question 795620: Hi,
I have trouble understanding how to factorise polynomials. Can you help me with the following:
4a^2 - 20ab^2 + 25b^4
and
x^4-81
Thanks!!!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! I'll do the first one to get you started.
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last coefficient is  .
Now multiply the first coefficient by the last coefficient to get  .
Now the question is: what two whole numbers multiply to  (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,2,4,5,10,20,25,50,100
-1,-2,-4,-5,-10,-20,-25,-50,-100
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*100 = 100
2*50 = 100
4*25 = 100
5*20 = 100
10*10 = 100
(-1)*(-100) = 100
(-2)*(-50) = 100
(-4)*(-25) = 100
(-5)*(-20) = 100
(-10)*(-10) = 100
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | 100 | 1+100=101 | | 2 | 50 | 2+50=52 | | 4 | 25 | 4+25=29 | | 5 | 20 | 5+20=25 | | 10 | 10 | 10+10=20 | | -1 | -100 | -1+(-100)=-101 | | -2 | -50 | -2+(-50)=-52 | | -4 | -25 | -4+(-25)=-29 | | -5 | -20 | -5+(-20)=-25 | | -10 | -10 | -10+(-10)=-20 |
From the table, we can see that the two numbers  and add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out  from the second group.
 Factor out 
 Condense
Therefore, factors to
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