Expand and simplify $x(x+2)(x+3)$.
To expand the expression \(x(x+2)(x+3)\), we recognize that this is a product of three factors. We can simplify the process by first calculating the product of the two binomials and then distributing the single variable \(x\) across the resulting trinomial. The distributive property states that \(a(b+c) = ab + ac\), and we can apply this alongside the FOIL method, which helps us multiply two binomials of the form \((x+a)(x+b) = x^2 + (a+b)x + ab\).
We first multiply the two binomials \((x+2)\) and \((x+3)\). Using the distributive property, we calculate:
\[(x+2)(x+3) = x(x) + x(3) + 2(x) + 2(3)\]
Simplifying the terms, we get:
\[(x+2)(x+3) = x^2 + 3x + 2x + 6\]
Combining the like terms \(3x\) and \(2x\), we have:
\[x^2 + 5x + 6\]
Now, we multiply the resulting quadratic expression by the remaining factor \(x\):
\[x(x^2 + 5x + 6)\]
Distributing \(x\) to each term inside the parentheses, we get:
\[x \cdot x^2 + x \cdot 5x + x \cdot 6\]
Performing these multiplications leads to:
\[x^3 + 5x^2 + 6x\]
This is the fully expanded and simplified form of the original expression.
Expanding algebraic expressions is a foundational skill used in physics and engineering to model physical systems. For example, if you are calculating the volume of a box with dimensions \(x\), \((x+2)\), and \((x+3)\) units, the volume is represented by the expression \(V = x(x+2)(x+3)\). By expanding this expression to \(V = x^3 + 5x^2 + 6x\), you can more easily analyze how the volume changes as the base dimension \(x\) increases. This polynomial representation is essential for solving optimization problems or graphing the behavior of the volume.