Final answer:
The question involves rewriting a cubic polynomial function using long division, understanding exponent rules, and solving quadratic equations using methods like the quadratic formula.
Step-by-step explanation:
The question pertains to rewriting a cubic polynomial function P(x) = x³ + 10x² - 23x - 132, given that (x+11) is a factor, using long division. The conceptual foundation lies in understanding the rules of exponents, which include the power of a product, the power of a quotient, and negative exponents. Notably, exponents can also be non-integer rational numbers, and their operations are well-defined and can be computed using a calculator. For example, xpxq = x(p+q) is a rule that applies here. To solve for the remaining factor, given that (x+11) is a factor of P(x), we would divide P(x) by (x+11) using long division or synthetic division to obtain a quadratic function, the product of which with (x+11) would yield the original cubic function P(x).
The discussion also touches on inverse operations, indicating that negative exponents denote reciprocals as in x-n = 1/xn. Particularly relevant is the concept that when we solve a quadratic equation, such as ax² + bx + c = 0, we often utilize the quadratic formula. This formula enables the determination of the possible values of x that satisfy the equation, by considering the coefficients of the quadratic expression. Similarly, in physics and other sciences, the manipulation of exponential notation and understanding the effects of operations on exponents is crucial, for instance, when dealing with very large or small numbers in scientific notation. These mathematical concepts form the foundation for solving a wide range of problems within various fields, reinforcing that mathematics offers multiple methods to arrive at the same solution, which can be employed to verify and confirm results.