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.
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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.
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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.
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