Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. 2x³+x+2; x+1

Verified step by step guidance

1. Identify the polynomials involved: the first polynomial is \(2x^{3} + x + 2\) and the second polynomial is \(x + 1\).
2. Rewrite the divisor \(x + 1\) in the form \(x - r\) to find the value of \(r\). Since \(x + 1 = x - (-1)\), we have \(r = -1\).
3. Apply the Factor Theorem by evaluating the first polynomial at \(x = r = -1\). Substitute \(-1\) into \(2x^{3} + x + 2\) to check if the result is zero.
4. If the result is zero, then \(x + 1\) is a factor of the first polynomial. If not, proceed to use synthetic division to divide \(2x^{3} + x + 2\) by \(x + 1\).
5. Set up synthetic division using \(r = -1\) and the coefficients of the first polynomial (note the missing \(x^{2}\) term has coefficient 0). Perform the division step-by-step to find the remainder. If the remainder is zero, \(x + 1\) is a factor; otherwise, it is not.

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. To check if a binomial like (x + 1) is a factor, substitute -1 into the polynomial and see if the result is zero.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form (x - c). It simplifies the division process by using only the coefficients, making it easier to find remainders and verify factors.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. Determining if one polynomial divides another without remainder helps break down complex expressions into simpler components.