Solve (2p^2-2p+2)^2 ... I got 2p^4-2p^2+4... Wrong

Question

the answer was 4p^4-8p^3+12p^2-8p+4, Im wondering how someone would get that answer.

Answer 1

Randall,

 

You did two things incorrectly.  

Number1: When you squared 2p you got 2p².  It should be 2²p² or 4p².

Number2: When you have to square of the sum of three terms, like (a+b+c)². you need to cross-multiply to get the correct answer which would be a² + ab + ac + ba + b² + bc + ca + cb +c² = a² + b² + c² + 2ab + 2ac + 2bc .  Instead of this answer, it appears that you got just a² + b² + c² and missed the three middle terms.  

 

Here is the correct solution:

 

(2p²-2p+2)² = 4(p²-p+1)² =

 

Step1:  Cross-multiply p² - p +1 with itself to get     (p²-p+1)² =  p⁴ -p³ +p² -p³ +p² -p +p² -p +1

 

So the full expression = 4(p⁴ -p³ +p²  -p³ +p² -p +p² -p +1)

 

Step2:  Collect terms to get   = 4(p⁴ -2p³ +3p² -2p + 1)

 

======================

 Just to check the answer, let's try it with p = 2 and p = 5:

 

p            (2p²-2p+2)²            4(p4 -2p3 +3p2 -2p + 1)

2                  36                             36

5               1764                          1764

 

Answer 2

 (2p^2 - 2p + 2)^2

 

= (2p^2 - 2p + 2)(2p^2 - 2p + 2)

 

Use the Distributive Property, not FOIL:

 

= 2p^2 (2p^2 - 2p + 2)

     - 2p (2p^2 - 2p + 2)

      + 2 (2p^2 - 2p + 2)

 

= (2p^2)(2p^2) - 2p(2p^2) + 2(2p^2)

- 2p(2p^2) - 2p(-2p) + 2(-2p)

+ 2(2p^2) + 2(-2p) + 2(2)

 

= 4p^4 - 4p^3 + 4p^2

            - 4p^3 + 4p^2 - 4p

                       + 4p^2 - 4p + 4

 

= 4p^4 - 8p^3 + 12p^2 - 8p + 4