Power functions, linear functions and quadratic functions are special cases of so-called polynomial functions. The general form of a polynomial function in the variable \(x\) is: \[f(x)=a_nx^n+a_{n-1}x^{n-1} + \cdots + a_2x^2+a_1x+a_0,\] with \(a_n\neq 0\).
The expression \[a_nx^n+a_{n-1}x^{n-1} + \cdots + a_2x^2+a_1x+a_0,\] with \(a_n\neq 0\) is called a polynomial. The parameters \(a_0, a_1, \ldots, a_n\) are called the coefficients of the polynomial. The highest exponent \(n\) is called the degree of the polynomial (hence linear and quadratic functions are also referred to as first degree functions and second degree functions, respectively). The term \(a_nx^n\) called leading term of the polynomial (whereby the highest coefficient \(a_n\) is not zero because otherwise you could have omitted this term).
The degree of \(f(x)=4x^3-3x^2+1\) is 3; the leading term is \(4x^3\) and the leading coefficient is \(4\).
The degree of \(g(x)=\tfrac{1}{2}x^4-1\) is 4; the leading term is \(\tfrac{1}{2}x^4\) and the leading coefficient is \(\tfrac{1}{2}\).
The degree of \(h(x)= x^5-5x^3-x^2+4x+1\) is 5; the leading term is \(x^5\) and the leading coefficient is \(1\).
The polynomial \[a_nx^n+a_{n-1}x^{n-1} + \cdots + a_2x^2+a_1x+a_0\] has the following short notation via the summation symbol: \[\sum_{k=0}^{n}a_kx^k\]
Linear and quadratic functions are polynomial functions of degree 1 and 2.
Power functions are polynomials functions with only one term.
You have alrteady seen that real quadratic functions can be factored as \(f(x)=x^2-3x+2=(x-1)(x-2)\). This also happens for higher degree polynomials. Expansion of brackets leads then to the standard form.
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The function \[f(x) = x^5-5x^3-x^2+4x+1\] whose graph over the interval \((-2.4, -2.4)\) is shown below is an example of a polynomial of degree 5 with leading term \(x^5\).
JYou clearly see that the graph has 4 extrema and 5 zeros. In general it is true that the graph of a polynomial function of degree \(n\) has\(n-1\) extrema and \(n\) zeros (at least when you ount with multiplicities, otherwise there can be less zeros).
Polynomial functions of degree three and higher
