The cubic expansion of a plus b plus c represents the volume of a cube with side length a plus b plus c. When we expand this algebraic expression, we get multiple terms: a cubed, b cubed, c cubed, plus various cross terms like 3 a squared b, and finally the triple product 6 a b c. Each term in this expansion has a specific geometric interpretation as we'll explore.
Today we'll explore the expansion of a plus b plus c to the third power. When we expand this expression, we get ten terms: three pure cube terms, six square terms, and one triple product term with coefficient six.
Now let's visualize this geometrically. A cube with side length a plus b plus c can be decomposed into twenty-seven smaller rectangular pieces. We can see how the segments of length a, b, and c combine to form the total side length. The internal grid lines show how the cube is divided, with each division corresponding to the boundaries between a, b, and c segments. This three-dimensional structure helps us understand why the algebraic expansion has so many terms.
Each small cube in our three by three by three grid corresponds to a term in the algebraic expansion. The pure cube terms like a cubed appear three times, the square terms like three a squared b appear six times with various combinations, and the triple product six a b c appears once. This geometric visualization makes it clear why we get exactly these coefficients and terms in the expansion.
Let's analyze each type of term systematically. In our twenty-seven piece cube, we have three cubic terms shown in red: a cubed, b cubed, and c cubed. We have eighteen rectangular prism terms shown in orange and purple, which include terms like three a squared b, three a squared c, and so on. Finally, we have six rectangular box terms shown in yellow, all contributing to the coefficient six a b c. Each color represents a different geometric volume type in our cube decomposition.
Now let's derive this expansion systematically using algebra. We start by writing the cube as the product of a plus b plus c times the square of a plus b plus c. First, we expand the square to get a squared plus b squared plus c squared plus two a b plus two a c plus two b c. Then we multiply this entire expression by a plus b plus c using the distributive property. We distribute each term carefully, collecting like terms to arrive at our final expansion with the correct coefficients.
Let's verify our expansion with a concrete example. Using a equals two, b equals three, and c equals one, we calculate directly: two plus three plus one cubed equals six cubed, which is two hundred sixteen. Now using our expansion formula, we compute each term: two cubed is eight, three cubed is twenty-seven, one cubed is one. The square terms give us thirty-six, twelve, fifty-four, twenty-seven, six, and nine. The triple product term gives us thirty-six. Adding all terms: eight plus twenty-seven plus one plus thirty-six plus twelve plus fifty-four plus twenty-seven plus six plus nine plus thirty-six equals exactly two hundred sixteen, confirming our formula works perfectly.