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"markdown": "# Polynomials\n\n\n\nIn this section we use the following add-on packages:\n\n``` {.julia .cell-code}\nusing SymPy\nusing Plots\n```\n\n\n\n\n---\n\n\nPolynomials are a particular class of expressions that are simple enough to have many properties that can be analyzed. In particular, the key concepts of calculus: limits, continuity, derivatives, and integrals are all relatively trivial for polynomial functions. However, polynomials are flexible enough that they can be used to approximate a wide variety of functions. Indeed, though we don't pursue this, we mention that `Julia`'s `ApproxFun` package exploits this to great advantage.\n\n\nHere we discuss some vocabulary and basic facts related to polynomials and show how the add-on `SymPy` package can be used to model polynomial expressions within `SymPy`. `SymPy` provides a Computer Algebra System (CAS) for `Julia`. In this case, by leveraging a mature `Python` package [SymPy](https://www.sympy.org/). Later we will discuss the `Polynomials` package for polynomials.\n\n\nFor our purposes, a *monomial* is simply a non-negative integer power of $x$ (or some other indeterminate symbol) possibly multiplied by a scalar constant. For example, $5x^4$ is a monomial, as are constants, such as $-2=-2x^0$ and the symbol itself, as $x = x^1$. In general, one may consider restrictions on where the constants can come from, and consider more than one symbol, but we won't pursue this here, restricting ourselves to the case of a single variable and real coefficients.\n\n\nA *polynomial* is a sum of monomials. After combining terms with same powers, a non-zero polynomial may be written uniquely as:\n\n\n\n$$\na_n x^n + a_{n-1}x^{n-1} + \\cdots a_1 x + a_0, \\quad a_n \\neq 0\n$$\n\n::: {.cell cache='true' hold='true' execution_count=4}\n\n::: {.cell-output .cell-output-display execution_count=5}\n```{=html}\n<div class=\"d-flex justify-content-center\"> <figure> <img src=\"data:image/gif;base64,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 truncated
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Reference in new issue
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