explain (a+b+c)^3 graphically and mathematically. i want you to explain slowly with good examples

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Welcome to our exploration of cubic expansion. Today we'll learn about the formula (a+b+c) cubed, which is a fundamental concept in algebra. When we expand this expression, we get ten different terms: a cubed, b cubed, c cubed, plus six mixed terms with coefficients of 3, and finally 6abc. This expansion has many practical applications in mathematics and science, from calculating volumes to solving complex polynomial equations. Now let's understand the cubic expansion through step-by-step multiplication. We start with (a+b+c) cubed, which equals (a+b+c) times (a+b+c) times (a+b+c). First, we multiply the first two factors to get a squared plus b squared plus c squared plus 2ab plus 2ac plus 2bc. Then we multiply this result by the third factor (a+b+c). When we distribute each term, we get the cubic terms a cubed, b cubed, and c cubed in blue. The quadratic terms combine to give us 3a squared b, 3a squared c, and so on, shown in red. Finally, the mixed terms combine to give us 6abc in green. This systematic approach shows us exactly how each term in the final expansion is formed. Now let's visualize the cubic expansion geometrically using a three-dimensional cube. When we have (a+b+c) cubed, we can think of this as the volume of a cube with side length (a+b+c). If we divide this large cube into smaller pieces based on the lengths a, b, and c, each piece corresponds exactly to a term in our algebraic expansion. The blue, red, and green corner cubes represent a cubed, b cubed, and c cubed respectively. The yellow and orange rectangular prisms represent terms like 3a squared b and 3a squared c. The purple piece represents the 6abc term. This geometric visualization helps us understand why we get exactly these coefficients in the expansion - they correspond to the number of identical pieces of each type when we systematically divide the cube. Let's work through a concrete numerical example to verify our expansion formula. We'll calculate (2+3+1) cubed using both direct calculation and the expansion method. First, the direct approach: 2 plus 3 plus 1 equals 6, and 6 cubed equals 216. Now using our expansion formula with a equals 2, b equals 3, and c equals 1. We calculate each term: 2 cubed equals 8, 3 cubed equals 27, 1 cubed equals 1. For the mixed terms: 3 times 2 squared times 3 equals 36, 3 times 2 squared times 1 equals 12, and so on. Finally, 6 times 2 times 3 times 1 equals 36. Now let's add them step by step: 8 plus 27 plus 1 equals 36, then 36 plus 36 equals 72, continuing this process until we reach 216. This confirms our expansion formula is correct! Let's explore the patterns in our cubic expansion and see its practical applications. The coefficients follow a clear pattern: 1 for the pure cubic terms, 3 for the mixed quadratic terms, and 6 for the triple product term. These numbers have combinatorial meaning - coefficient 3 represents choosing 2 positions for the same variable from 3 slots, while coefficient 6 equals 3 factorial, representing all ways to arrange three different variables. This expansion has many real-world applications: calculating volumes in engineering, expanding polynomials in physics, and performing algebraic manipulations in calculus. For example, (x+2y+3z) cubed expands to include terms with the original coefficients multiplied by the powers of the constants. In summary, we've learned four key concepts: the algebraic expansion method, geometric cube interpretation, coefficient pattern recognition, and practical applications. These tools will help you tackle similar polynomial expansions with confidence.