This problem has four parts, Part A, Part B, Part C, and Part D. Read the proof, and then answer all four 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 a power as a product of the base
3. _[blank 1]_ Distributive Property
4. a2−ab−ab+b2 Distributive Property
5. a2−2ab+b2 _[blank 2]_










Part A: Which statement correctly fills in blank 1 to complete the proof?

Part B: Which reason correctly fills in blank 2 to complete the proof?

Part C: Why is Statement 3 valid?

Part D: Why is Reason 5 valid?

A: a(a−b)−a(a−b)
D: In order to simplify the expression in the previous step, the Commutative Property of Addition must be applied.
C: In order to multiply the binomials in the previous step, one binomial must be distributed to each term in the other binomial.
A: a(a−b)−b(a−b)
C: In order to multiply the binomials in the previous step, the like terms must be combined.
A: a(a+b)−b(a+b)
D: In order to simplify the expression in the previous step, the Distributive Property must be applied.
D: In order to simplify the expression in the previous step, the like terms must be combined.
B: Combine like terms.
B: Commutative Property of Addition
B: Distributive Property

Much ado about nothing

(a-b)^2
= (a-b)(a-b)
= a^2 - ab - ab + b^2
= a^2 - 2ab + b^2

Part A: The correct statement to fill in blank 1 is "a(a−b)−a(a−b)". This step involves distributing the binomial (a−b) to both terms inside the other binomial.

Part B: The correct reason to fill in blank 2 is "Combine like terms". This step involves simplifying the expression by combining the like terms (-ab and -ab) to get -2ab.

Part C: Statement 3 is valid because of the Distributive Property. The Distributive Property states that when you multiply a sum by a number, you can distribute the multiplication to each term inside the sum. In this case, (a−b)(a−b) can be expanded using the Distributive Property to get a(a−b)−b(a−b).

Part D: Reason 5 is valid because it involves simplifying the expression by combining the like terms (-ab and -ab) to get -2ab. This step follows from Step 4, where each term in the expanded expression is written separately before combining the like terms.

So, the correct answers are:

Part A: a(a−b)−a(a−b)
Part B: Combine like terms
Part C: In order to multiply the binomials in the previous step, one binomial must be distributed to each term in the other binomial.
Part D: In order to simplify the expression in the previous step, the like terms must be combined.