Here are the essential concepts you must grasp in order to answer the question correctly.
Synthetic Division
Synthetic division is a simplified form of polynomial long division that allows for the efficient division of a polynomial by a linear factor of the form (x - c). It involves using the coefficients of the polynomial and performing a series of multiplications and additions to find the quotient and remainder. This method is particularly useful for evaluating polynomials at specific values, such as finding ƒ(2) in this case.
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Polynomial Evaluation
Polynomial evaluation is the process of calculating the value of a polynomial function at a specific input. For a polynomial ƒ(x), substituting a value for x allows us to determine the output of the function. In this question, we are tasked with evaluating the polynomial ƒ(x) = x^5 + 4x^2 - 2x - 4 at x = 2, which can be efficiently done using synthetic division.
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Introduction to Polynomials
Remainder Theorem
The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is equal to f(c). This theorem provides a quick way to evaluate polynomials at specific points without fully performing the division. In this context, using synthetic division to find ƒ(2) will yield the remainder, which directly gives us the value of the polynomial at that point.
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