What is the result of dividing $x^3-4$ by $x+2$ ?
The problem asks us to divide the polynomial \(x^3 - 4\) by the linear binomial \(x + 2\). We aim to find the result in the form \(Q(x) + \frac{R}{x + 2}\), where \(Q(x)\) is the quotient and \(R\) is a constant remainder. This is a classic polynomial division problem that can be solved using either long division or algebraic manipulation.
We can solve this by rewriting the numerator \(x^3 - 4\) in a way that allows us to factor out the divisor \(x + 2\). A useful identity is the sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). If we let \(a = x\) and \(b = 2\), then \(x^3 + 8 = (x + 2)(x^2 - 2x + 4)\). We can write our original numerator as:
\[x^3 - 4 = x^3 + 8 - 12\]
Now, we can perform the division:
\[\frac{x^3 - 4}{x + 2} = \frac{x^3 + 8 - 12}{x + 2} = \frac{x^3 + 8}{x + 2} - \frac{12}{x + 2}\]
Substituting the sum of cubes factorization, we get:
\[\frac{x^3 + 8}{x + 2} - \frac{12}{x + 2} = \frac{(x + 2)(x^2 - 2x + 4)}{x + 2} - \frac{12}{x + 2} = x^2 - 2x + 4 - \frac{12}{x + 2}\]
Comparing our calculated result \(x^2 - 2x + 4 - \frac{12}{x + 2}\) with the given choices, we see that it matches the second option. Thus, the quotient is \(x^2 - 2x + 4\) and the remainder is \(-12\). The complete result is \(\boxed{x^2 - 2x + 4 - \frac{12}{x + 2}}\).
Polynomial division is widely used in fields like control theory and signal processing, where it helps in simplifying transfer functions or analyzing the behavior of systems described by differential equations. For instance, if you are designing a digital filter, you often need to divide polynomials to find the impulse response or to stabilize the system's output. Understanding these algebraic techniques allows engineers and mathematicians to decompose complex expressions into manageable components, making it easier to solve for specific system parameters.