What is the completely factored form of $x^3-64 x$?
The objective is to find the completely factored form of the cubic polynomial \(x^3 - 64x\). We begin by looking for a greatest common factor (GCF) that divides all terms of the polynomial. In the expression \(x^3 - 64x\), the variable \(x\) is common to both \(x^3\) and \(-64x\). By factoring out \(x\), we can rewrite the expression as \(x(x^2 - 64)\).
Next, we focus on the binomial inside the parentheses, which is \(x^2 - 64\). This expression is a difference of two squares because it is of the form \(a^2 - b^2\), where \(a = x\) and \(b = 8\) (since \(8^2 = 64\)). The difference of squares formula is given by:
\[a^2 - b^2 = (a - b)(a + b).\]
Applying this identity to \(x^2 - 8^2\), we get \((x - 8)(x + 8)\).
Now, we combine the factored common factor \(x\) from the first step with the factors obtained from the difference of squares in the second step. This gives us the complete factorization:
\[x(x - 8)(x + 8).\]
Comparing this result to the given options:
The result matches option (a).
Factoring polynomials is a fundamental skill in algebra with many practical applications. For instance, in engineering and physics, factoring helps determine the roots of equations, which can represent points where a system reaches equilibrium, crosses an axis, or changes state. When modeling the trajectory of a projectile, the factored form of the equation allows us to easily identify the times when the object is on the ground (setting the height function to zero). Understanding these algebraic techniques is essential for solving more complex design and analysis problems in science and technology.