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

Verified step by step guidance

1. Identify the divisor from the second polynomial, which is \(x + 1\). According to the factor theorem, if \(x + 1\) is a factor of the first polynomial, then substituting \(-1\) into the first polynomial should yield zero.
2. Set up synthetic division using \(-1\) as the divisor. Write down the coefficients of the first polynomial \(2x^3 + 0x^2 + x + 2\), which are \([2, 0, 1, 2]\).
3. Bring down the leading coefficient \(2\) to the bottom row.
4. Multiply \(-1\) by the number just written on the bottom row (\(2\)) and write the result under the next coefficient (\(0\)). Add this result to the coefficient above it and write the sum below.
5. Repeat the multiplication and addition process for the remaining coefficients. If the final number on the bottom row (the remainder) is zero, then \(x + 1\) is a factor of the polynomial.

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 the root of the second polynomial, which is -1, to check if it equals zero.

Synthetic Division

Synthetic division is a simplified form of polynomial long division that allows for the division of a polynomial by a linear factor of the form (x - c). It is a more efficient method for performing polynomial division, especially when determining factors. In this case, we will use synthetic division to divide the first polynomial by (x + 1) to see if the remainder is zero, confirming that (x + 1) is a factor.

Polynomial Remainder Theorem

The Polynomial Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor (x - c), the remainder of this division is f(c). This theorem is crucial for understanding the results of synthetic division. If the remainder is zero, it indicates that the divisor is indeed a factor of the polynomial, which is the goal of the problem at hand.