1. Home
  2. Math Notes
  3. Calculus I
  4. Common Functions

Polynomial Function

Polynomial is a function of the form y=f(x)=anxn+an−1xn−1+…+a2x2+a1x+a0{y}={f{{\left({x}\right)}}}={a}_{{n}}{{x}}^{{n}}+{a}_{{{n}-{1}}}{{x}}^{{{n}-{1}}}+\ldots+{a}_{{2}}{{x}}^{{2}}+{a}_{{1}}{x}+{a}_{{0}}, where n is nonnegative integer and a0, a1, …, an−1, an{a}_{{0}},\ {a}_{{1}},\ \ldots,\ {a}_{{{n}-{1}}},\ {a}_{{n}} are constants which are called coefficients of polynomial.

Domain of any polynomial is (−∞,∞){\left(-\infty,\infty\right)}.

If an≠0{a}_{{n}}\ne{0} then n{n} is the degree of polynomial. For example, f(x)=−2x5+3x+3{f{{\left({x}\right)}}}=-{2}{{x}}^{{5}}+{3}{x}+\sqrt{{{3}}} is a polynomial of degree 5.

If degree of polynomial is n=1, then f(x)=mx+b{\color{blue}{{{f{{\left({x}\right)}}}={m}{x}+{b}}}} and it is linear function.

In case degree is 2 then f(x)=ax2+bx+c{\color{red}{{{f{{\left({x}\right)}}}={a}{{x}}^{{2}}+{b}{x}+{c}}}} and this function is called quadratic.

A polynomial of degree 3 f(x)=ax3+bx2+cx+d{\color{green}{{{f{{\left({x}\right)}}}={a}{{x}}^{{3}}+{b}{{x}}^{{2}}+{c}{x}+{d}}}} is called cubic function.

Graph of polynomial functions have different shapes. On the figure you can see graph of polynomials with degree n=2,3,4{n}={2},{3},{4}.

polynomial function