Question

how to factor this v^3+4V^2+3V

Answer 1

Equation at the end of step  1  : ((v3) + 22v2) + 3v 
Pulling out like terms :

 3.1     Pull out like factors :

   v3 + 4v2 + 3v  =   v • (v2 + 4v + 3) 

Trying to factor by splitting the middle term

 3.2     Factoring  v2 + 4v + 3 

The first term is,  v2  its coefficient is  1 .
The middle term is,  +4v  its coefficient is  4 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 3 = 3 

Step-2 : Find two factors of  3  whose sum equals the coefficient of the middle term, which is   4 .

     -3   +   -1   =   -4     -1   +   -3   =   -4     1   +   3   =   4   That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  1  and  3 
                     v2 + 1v + 3v + 3

Step-4 : Add up the first 2 terms, pulling out like factors :
                    v • (v+1)
              Add up the last 2 terms, pulling out common factors :
                    3 • (v+1)
Step-5 : Add up the four terms of step 4 :
                    (v+3)  •  (v+1)
             Which is the desired factorization

Final result : v • (v + 3) • (v + 1)       

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