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

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Step 1: Use the Factor Theorem, which states that \( x + 1 \) is a factor of \( f(x) \) if and only if \( f(-1) = 0 \).

Step 2: Substitute \( x = -1 \) into the polynomial \( f(x) = x^3 + 2x^2 + 3x + 2 \).

Step 3: Calculate \( f(-1) = (-1)^3 + 2(-1)^2 + 3(-1) + 2 \).

Step 4: Simplify the expression: \( (-1)^3 = -1 \), \( 2(-1)^2 = 2 \), \( 3(-1) = -3 \), and \( +2 \).

Step 5: Add the results: \( -1 + 2 - 3 + 2 \) and check if the sum equals zero to determine if \( x + 1 \) is a factor.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, ƒ(x) = x^3 + 2x^2 + 3x + 2 is a polynomial of degree 3. Understanding polynomial functions is essential for analyzing their behavior, including factors and roots.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, which can be simpler polynomials. To determine if x + 1 is a factor of ƒ(x), one can use polynomial long division or synthetic division. If the remainder is zero, then x + 1 is indeed a factor.

Remainder Theorem

The Remainder Theorem states that for a polynomial f(x), if you divide it by (x - c), the remainder of this division is f(c). This theorem can be used to quickly check if x + 1 is a factor of ƒ(x) by evaluating ƒ(-1). If ƒ(-1) equals zero, then x + 1 is a factor.