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Now, let's break this down further. **Horner's method** works by iteratively calculating intermediate results. Starting from the highest degree coefficient, we multiply the current result by the input value (let's call it 'x₀') and add the next coefficient. We then repeat this process until we reach the constant term. This approach significantly reduces the number of multiplications needed, making it computationally advantageous. The efficiency gains become particularly noticeable as the degree of the polynomial increases. This method isn't just about speed, though; it also helps to minimize the accumulation of rounding errors, which can be a problem with direct evaluation methods. The nested structure of **Horner's method** helps to keep the numbers within a more manageable range, improving the accuracy of the result. So, in essence, Horner's method provides a more stable and faster way to evaluate polynomials.
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