Operations with Polynomials..?

Question

Mr. Murphy's math class has been studying operations with polynomials and factoring. Today in class Mr. Murphy wrote the following pattern on the board:

 

1*2*3=4*1^2+2*1

 

2*3*4=5*2^2+2*2

 

3*4*5=6*3^2+2*3

 

4*5*6=7*4^2+2*4

 

He then had the class verify that each equation in the pattern is true. He told the class that the pattern illustrates a true statement about any three consecutive integers. He had the students test this claim by determining and checking the next two equations in the pattern.

 

5*6*7=8*5^2+2*5

 

6*7*8=9*6^2+2*6

 

Mr. Murphy then asked the class to determine the rule for this pattern. Write a statement in words that describes the pattern. Prove the statement. 

Answer 1

n(n+1)(n+2) = (n+3)(n)² + 2n

 

Proof:  (n+3)n² + 2n = n[n(n + 3) + 2]

 

                              = n(n² + 3n + 2)

 

                              = n(n + 1)(n + 2)