Welcome to our exploration of the cube of the sum of two variables. Today we will learn how to expand the expression a plus b to the third power. This is a fundamental algebraic identity that we'll derive step by step using multiplication and distribution.
The first step in expanding a plus b cubed is to rewrite it as a product. We can express a plus b to the third power as a plus b times a plus b squared. This approach breaks down the complex cube into simpler components that are easier to work with.
In step two, we expand the square term. We know that a plus b squared equals a squared plus two a b plus b squared. We substitute this expansion into our expression, giving us a plus b times the quantity a squared plus two a b plus b squared.
In step three, we distribute and multiply. We multiply a by each term in the second parenthesis, giving us a cubed plus two a squared b plus a b squared. Then we multiply b by each term, giving us b a squared plus two a b squared plus b cubed.
In the final step, we combine like terms. We have two a squared b terms that combine to give three a squared b, and two a b squared terms that combine to give three a b squared. This gives us our final answer: a plus b cubed equals a cubed plus three a squared b plus three a b squared plus b cubed.