What is the justification for the first step in proving the formula for factoring the sum of cubes?
distributive property commutative property definition of additive inverse definition of multiplicative inverse
The Correct Answer and Explanation is:
The justification for the first step in proving the formula for factoring the sum of cubes,
a3+b3=(a+b)(a2−ab+b2)a3+b3=(a+b)(a2−ab+b2), is the distributive property.
A mathematical proof or verification of this formula typically begins with the factored, more complex side of the equation,
(a+b)(a2−ab+b2)(a+b)(a2−ab+b2), and demonstrates through algebraic manipulation that it simplifies to
a3+b3a3+b3.
The very first step in this process is to expand the product of the binomial
(a+b)(a+b)and the trinomial
(a2−ab+b2)(a2−ab+b2). The distributive property states that to multiply a sum by an expression, one can multiply each term of the sum by that expression individually. In this case, each term in the first parenthesis, namely a and b, must be distributed across the entire second parenthesis.
The first step is written as:
(a+b)(a2−ab+b2)=a(a2−ab+b2)+b(a2−ab+b2)(a+b)(a2−ab+b2)=a(a2−ab+b2)+b(a2−ab+b2)This action is a direct application of the distributive property. This property is the fundamental rule that allows for the expansion of factored expressions.
The subsequent steps also use the distributive property again (to multiply a into its parenthesis and b into its), followed by the commutative property to rearrange terms and the definition of the additive inverse to cancel out opposing terms like
−a2b−a2band
+a2b+a2b. However, the crucial, initiating step of the entire expansion relies solely on the distributive property. It is the principle that unlocks the multiplication of these two polynomial factors.thumb_upthumb_down
