What is the quotient of $\left(x^3+8\right) \div(x+2)$?
The expression given is a polynomial division of a sum of cubes, \(x^3 + 8\), by a linear term, \(x + 2\). The dividend can be written as \(x^3 + 2^3\), which is in the form \(a^3 + b^3\) where \(a = x\) and \(b = 2\). We aim to find the quotient \(\frac{x^3 + 8}{x + 2}\).
We can use the algebraic identity for the sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Setting \(a = x\) and \(b = 2\), the identity becomes:
\[x^3 + 2^3 = (x + 2)(x^2 - (x)(2) + 2^2)\]
\[x^3 + 8 = (x + 2)(x^2 - 2x + 4)\]
This factorization allows us to express the dividend as a product involving the divisor.
Now, we perform the division \((x^3 + 8) \div (x + 2)\):
\[\frac{x^3 + 8}{x + 2} = \frac{(x + 2)(x^2 - 2x + 4)}{x + 2}\]
Since \(x+2
eq 0\), we can cancel the term \((x+2)\) from the numerator and the denominator:
\[\frac{(x + 2)(x^2 - 2x + 4)}{x + 2} = x^2 - 2x + 4\]
The quotient is thus \(x^2 - 2x + 4\).
Comparing our result \(x^2 - 2x + 4\) to the provided options, we see that it matches the first option. The quotient of \((x^3+8) \div (x+2)\) is \(\boxed{x^2-2x+4}\).
Understanding polynomial division is essential in algebra for simplifying complex expressions and solving higher-degree equations. For example, in engineering, when designing a system that can be modeled by a polynomial function, engineers often use division to simplify the relationship between input and output variables, making it easier to analyze the system's behavior or optimize its performance. Factoring tools like the sum-of-cubes identity provide quick ways to reduce complexity in such real-world calculations.