In Exercises 91–100, find all values of x satisfying the given conditions. y = x^3 + 4x^2 - x + 6 and y = 10

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, the function y = x^3 + 4x^2 - x + 6 is a cubic polynomial, which means it has a degree of three. Understanding polynomial functions is essential for analyzing their behavior, including finding their roots or points of intersection with other functions.

Finding Intersections

Finding intersections between two functions involves determining the points where their graphs meet. This is done by setting the two equations equal to each other. In this problem, we set the polynomial function equal to the constant value of 10, leading to the equation x^3 + 4x^2 - x + 6 = 10, which simplifies to finding the roots of a new polynomial.

Root Finding Techniques

Root finding techniques are methods used to determine the values of x for which a polynomial equals zero. Common methods include factoring, using the Rational Root Theorem, synthetic division, and numerical methods like the Newton-Raphson method. In this context, once the equation is simplified, applying these techniques will help identify the values of x that satisfy the original condition.