SOLUTION: Factorize the following: a^3+3a^2+3a+2

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Question 1164256: Factorize the following: a^3+3a^2+3a+2
Found 3 solutions by ikleyn, greenestamps, Edwin McCravy:
Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
.

    a%5E3+%2B+3a%5E2+%2B+3a+%2B+2 = re-grouping = %28a%5E3+%2B+3a%5E2+%2B+3a+%2B+1%29 + 1 = %28a%2B1%29%5E3 + 1 = %28%28a%2B1%29+%2B+1%29%2A%28%28a%2B1%29%5E2+-+%28a%2B1%29%2A1+%2B+1%29


and you can simplify it further and complete.



On the way, I used these identities


    x%5E3+%2B+3x%5E2+%2B+3x+%2B+1 = %28x%2B1%29%5E3;

    x%5E3+%2B+1 = %28x%2B1%29%2A%28x%5E2+-+x+%2B+1%29,


which every student must know from his (or her) Algebra class.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The rational roots theorem tells us the only possible rational roots are 1, -1, 2, and -2.

Synthetic division shows us that -2 is a root:

  -2 | 1  3  3  2
     |   -2 -2 -2
     ------------
       1  1  1  0

ANSWER: a%5E3%2B3a%5E2%2B3a%2B2+=+%28a%2B2%29%28a%5E2%2Ba%2B1%29

Here is a different path to the answer; this is what first caught my eye when I saw the problem.

a%5E3%2B3a%5E2%2B3a%2B2+=+%28a%5E3%2B3a%5E2%2B3a%2B1%29%2B1+=+%28a%2B1%29%5E3%2B1%5E3

Use the factorization pattern x%5E3%2By%5E3+=+%28x%2By%29%28x%5E2-xy%2By%5E2%29:

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

The other method is fine if you have memorized the binomial expansion
identity:

(x + y)³ = x³ + 3x²y + 3xy² + y³

and observe how close it is to the given cubic polynomial.  But most
students probably haven't memorized this.  So I did it the other way:

--------------------------------------------------------------------

a%5E3%2B3a%5E2%2B3a%2B2

We search for rational roots which are ± factors of 2,
which are ±1, ±2

There are no sign changes so there are no positive roots,
So the only rational roots, if any, are -1 and -2

We try -1

-1 | 1  3  3  2
   |   -1 -2 -1 
     1  2  1  1

No, for the remainder is not 0.

We try -1

-2 | 1  3  3  2
   |   -2 -2 -2 
     1  1  1  0

Yes, for the remainder is 0.

We divided by a+2 and the quotient is determined by
the first three numbers on the bottom of the synthetic
division.  The quotient is 1a² + 1a + 1 or a² + a + 1.

(a + 2)(a² + a + 1)      <--answer

Edwin