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

Verified step by step guidance

1. Identify the divisor from the second polynomial, which is \(x + 3\). According to the factor theorem, if \(x + 3\) is a factor, then \(x = -3\) should be a root of the first polynomial.
2. Set up synthetic division using \(x = -3\) as the divisor. Write down the coefficients of the first polynomial \(2x^4 + 5x^3 - 2x^2 + 5x + 6\), which are \([2, 5, -2, 5, 6]\).
3. Perform synthetic division: Bring down the leading coefficient (2) to the bottom row. Multiply it by \(-3\) and add to the next coefficient (5). Continue this process for each coefficient.
4. Check the remainder after completing the synthetic division. If the remainder is 0, then \(x + 3\) is a factor of the polynomial.
5. Conclude whether \(x + 3\) is a factor based on the remainder. If the remainder is not 0, then \(x + 3\) is not a factor.

Key Concepts

Factor Theorem

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

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

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.