Solve each problem. Is x+1 a factor of ƒ(x)=x³+2x²+3x+2?

Verified step by step guidance

1. Recall the Factor Theorem, which states that if \(x - c\) is a factor of a polynomial \(f(x)\), then \(f(c) = 0\). In this problem, since the factor is \(x + 1\), rewrite it as \(x - (-1)\), so \(c = -1\).
2. Evaluate the polynomial \(f(x) = x^3 + 2x^2 + 3x + 2\) at \(x = -1\) by substituting \(-1\) into the polynomial: calculate \(f(-1) = (-1)^3 + 2(-1)^2 + 3(-1) + 2\).
3. Simplify each term in the expression: \((-1)^3 = -1\), \$2(-1)^2 = 2(1) = 2\(, \)3(-1) = -3\(, and the constant term is \)2$.
4. Add the simplified terms together: \(-1 + 2 - 3 + 2\) to find the value of \(f(-1)\).
5. If the result of \(f(-1)\) is zero, then \(x + 1\) is a factor of \(f(x)\). If it is not zero, then \(x + 1\) is not a factor.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding the structure of polynomials helps in analyzing their factors and roots.

Factor Theorem

The Factor Theorem states that (x - c) is a factor of a polynomial ƒ(x) if and only if ƒ(c) = 0. This theorem is used to test whether a given binomial is a factor by substituting the root into the polynomial.

Evaluating Polynomials

Evaluating a polynomial involves substituting a specific value for the variable and simplifying to find the result. This process is essential for applying the Factor Theorem to determine if a binomial is a factor.