Here are the essential concepts you must grasp in order to answer the question correctly.
Synthetic Division
Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form (x - k). It allows for quick calculations by using the coefficients of the polynomial and the value of k. This technique is particularly useful for determining if k is a root of the polynomial, as it provides the remainder directly, which indicates whether the polynomial evaluates to zero at that point.
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Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is ƒ(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a non-negative integer and a_n is not zero. Understanding the structure of polynomial functions is essential for analyzing their behavior, including finding zeros and evaluating the function at specific points.
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Evaluating a Function
Evaluating a function involves substituting a specific value for the variable in the function's expression to find the corresponding output. For example, to evaluate ƒ(k) for a given k, you replace x in the polynomial with k and compute the result. This process is crucial for determining whether k is a zero of the polynomial, as a zero means the function evaluates to zero at that point.
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