SOLUTION: Factor (a+4)^2 - 2(a+4) +1

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Question 238740: Factor
(a+4)^2 - 2(a+4) +1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let z=a%2B4. So the expression %28a%2B4%29%5E2+-+2%28a%2B4%29+%2B1 then becomes z%5E2-2z%2B1


Looking at the expression z%5E2-2z%2B1, we can see that the first coefficient is 1, the second coefficient is -2, and the last term is 1.


Now multiply the first coefficient 1 by the last term 1 to get %281%29%281%29=1.


Now the question is: what two whole numbers multiply to 1 (the previous product) and add to the second coefficient -2?


To find these two numbers, we need to list all of the factors of 1 (the previous product).


Factors of 1:
1
-1


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to 1.
1*1 = 1
(-1)*(-1) = 1

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -2:


First NumberSecond NumberSum
111+1=2
-1-1-1+(-1)=-2



From the table, we can see that the two numbers -1 and -1 add to -2 (the middle coefficient).


So the two numbers -1 and -1 both multiply to 1 and add to -2


Now replace the middle term -2z with -z-z. Remember, -1 and -1 add to -2. So this shows us that -z-z=-2z.


z%5E2%2Bhighlight%28-z-z%29%2B1 Replace the second term -2z with -z-z.


%28z%5E2-z%29%2B%28-z%2B1%29 Group the terms into two pairs.


z%28z-1%29%2B%28-z%2B1%29 Factor out the GCF z from the first group.


z%28z-1%29-1%28z-1%29 Factor out 1 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28z-1%29%28z-1%29 Combine like terms. Or factor out the common term z-1


%28z-1%29%5E2 Condense the terms.



So z%5E2-2z%2B1 factors to %28z-1%29%5E2.


In other words, z%5E2-2z%2B1=%28z-1%29%5E2.


Now plug in z=a%2B4 to go from %28z-1%29%5E2 to %28a%2B4-1%29%5E2. Now simplify to get %28a%2B3%29%5E2


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Answer:


So %28a%2B4%29%5E2+-+2%28a%2B4%29+%2B1 factors to %28a%2B3%29%5E2.


In other words, %28a%2B4%29%5E2+-+2%28a%2B4%29+%2B1=%28a%2B3%29%5E2.