Final Answer:
The polynomial P(x) = x³ + 8 can be factored as P(x) = (x + 2)(x² - 2x + 4).
Step-by-step explanation:
To factor the given polynomialP(x) = x³ + 8, we can use the sum of cubes formula, which states that \(a³ + b³ = (a + b)(a² - ab + b²)\). In this case, (a = x) and (b = 2). Applying the sum of cubes formula, we get:
P(x) = x³ + 2³ = (x + 2)(x² - 2x + 4)
So, P(x) can be factored as(x + 2)(x² - 2x + 4). This is the final factored form of the polynomial.
The factorization P(x) = (x + 2)(x² - 2x + 4)is obtained by factoring out the sum of cubes using the given polynomial. The factor (x + 2) represents the cube root term, and (x² - 2x + 4) represents the remaining quadratic factor. This quadratic factor does not factor further over the real numbers, so the factorization is considered complete.
The sum of cubes formula is a special case of factoring, often useful in simplifying expressions or solving equations involving cubic terms. Understanding such algebraic formulas can be crucial in various mathematical applications.