Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. 2x^4+5x^3-2x^2+5x+6; x+3

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. This theorem is essential for determining whether a given polynomial is a factor of another polynomial. In this context, we will evaluate the first polynomial at x = -3 to see if it equals zero, indicating that (x + 3) is a factor.

Synthetic division is a simplified method for dividing a polynomial by a linear factor of the form (x - c). It involves using the coefficients of the polynomial and performing a series of arithmetic operations to find the quotient and remainder. This technique is particularly useful for quickly determining if a polynomial is divisible by another without performing long division.

Polynomial functions are mathematical expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. Understanding the structure of polynomial functions, including their degree and leading coefficient, is crucial for analyzing their behavior, roots, and factors. In this problem, we are working with a fourth-degree polynomial, which influences the number of possible factors.