Welcome! Today we'll learn how to expand the expression a plus b to the third power. This is a fundamental algebraic skill. We can think of this geometrically as a cube with side length a plus b. Let's start by using the method where we write a plus b cubed as a plus b times a plus b squared.
Now let's expand a plus b squared. We use the formula x plus y squared equals x squared plus two x y plus y squared. Applying this to our expression, we get a plus b squared equals a squared plus two a b plus b squared. We can visualize this as a square divided into four parts: a squared, b squared, and two rectangles each with area a b.
Now we need to multiply a plus b by the expanded form a squared plus two a b plus b squared. We use the distributive property, multiplying each term in the first parentheses by each term in the second parentheses. This gives us a times the quantity a squared plus two a b plus b squared, plus b times the quantity a squared plus two a b plus b squared.
Now let's expand each distribution. First, a times the quantity a squared plus two a b plus b squared gives us a cubed plus two a squared b plus a b squared. Second, b times the quantity a squared plus two a b plus b squared gives us a squared b plus two a b squared plus b cubed. Adding these together and combining like terms, we get a cubed plus three a squared b plus three a b squared plus b cubed.
To summarize what we have learned: We successfully expanded a plus b cubed using algebraic distribution methods. The final formula is a plus b cubed equals a cubed plus three a squared b plus three a b squared plus b cubed. This follows the binomial theorem pattern with coefficients one, three, three, one from Pascal's triangle. This fundamental formula is widely used in algebra and calculus applications.