A calculus of the absurd
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4.3.4 Comparing coefficients
Would you like to know why this section went here? (I’m sure you wouldn’t, but I shall provide some context anyway). It goes here because it is useful for solving lots of problems that will occur later.
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Example 4.3.4 Let us suppose that we have these two polynomials
\(\seteqnumber{0}{4.}{30}\)
\begin{align}
& f(x) = ax^3 + bx^2 + cx + d, \\ & p(x) = 4x^3 + 3x^2 + 2x + 8.
\end{align}
And we know that \(f(x) = p(x)\). How would you find \(a, b, c\) and \(d\)?
This is not the hardest problem in the metaphorical book (far from it), but it’s also not a bad place to start. Intuitively we know that \(a = 4, b = 3, c = 2\) and \(d = 8\).
The general principle here is that two polynomial functions are equal if and only if their coefficients are equal1111 This seems obvious, but sometimes when we consider structures other than the real numbers two
polynomials can be equal, but their functions different. This is very confusing because when we are initially taught polynomials we are told that they are functions, but really it is better to think of a polynomial of a list of coefficients, which can be interpreted as a function..
In non-mathematics speak, this means that we look at the values in front of the numbers and check if they are the same.
TODO: some examples