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).