Polynomials

A polynomial in standard form is a function of the form

\[P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_0\]

where \(a_n\neq 0\). The natural number \(n\) is called the ‘degree’ of the polynomial. The real numbers \(a_n,a_{n-1},\cdots,a_1,a_0\) are called ‘coefficients’. The coefficient \(a_n\) is called the ‘leading coefficient’ and \(a_0\) the ‘constant term’. A polynomial in factored form (which does not always exist in real numbers) is a function of the form

\[P(x)=a(x-r_1)(x-r_2)\cdots(x-r_n)\]

where \(a\) is the leading coefficient and \(r_1,r_2,\dots r_n\) are the ‘roots’ (zeros, \(\boldsymbol x\)-intercepts) of the polynomial. A constant function \(f(x)=k\) is a polynomial of degree zero for every \(k\in\mathbb{R}\) such that \(k\neq 0\). The function \(f(x)=0\) is a polynomial of an undefined degree.