Question

This problem has two parts, Part A and Part B. Read the proof, and then answer both parts.

Given: (a+b)2

Prove (a+b)2=a2+2ab+b2.

Statements Reasons
1. (a+b)2 Given
2. (a+b)(a+b) Rewrite the power as a product of the base.
3. _[blank ]_ Distributive Property
4. a2+ab+ab+b2 Distributive Property
5. a2+2ab+b2 Combine like terms.










Part A: Which statement correctly fills the blank in Statement 3 to complete the proof?

Part B: Which analysis of the proof correctly describes the efficiency of steps?

Select one answer for Part A and select one answer for Part B.

A: a(a+b)+b(a+b)=a2+2ab+b2
A: b(a+b)+b(a+b)=a2+2ab+b2
A: 2a+2b+2ab=a2+2ab+b2
A: a2+ab−ab+b2=a2+2ab+b2
B: Using the distributive property first was unnecessary in Step 3 as it was undone in Step 4.
B: By focusing on one side of the equation, the process is the most direct path to finalize the proof.
B: The proof is not efficient because like terms are combined twice.

Answer 1

A: A(A+B)+B(A+B)=A^2+B^2+2AB