The function \(f(x) = x^3 - 1\) is an example of a third degree function in \(x\) with leading term \(x^3\). This polynomial can be written differently: \[x^3-1 = (x-1)(x^2+x+1)\] Check this by expanding brackets. The factor \(x-1\) does not come completely out of the blue, but is a consequence of the fact that \(x=1\) is a root of the polynomial, i.e., \(f(1)=0\). More generally, the following factor theorem holds:

We do not discuss here the method of finding such a polynomial function \(g(x)\) given a factor \(x-a\), except that we tell you that it works with the so-called long division of polynomials (similar to long division with numbers) and that in simple cases the polynomial function \(g(x)\) can be computed by a direct method..

We consider the cubic polynomial\(x^3-7x-6\).
By looking carefully at the polynomial it can be noted that \(x=-1\) is a root of the polynomial. This means that \(x+1\) is a factor of the factorisation of the cubic polynomial and that \[x^3-7x-6=(x+1) (ax^2+bx+c)\] for some numbers \(a\), \(b\) and \(c\). Removal of brackets in the right-hand side and collection of terms gives \[x^3-7x-6=ax^3+(a+b)x^2+(b+c)x+c\] Now we compare the coefficients of like terms on the left- and right-hand side annd find that \[a=1,\quad a+b=0,\quad b+c=-7,\quad c=-6\] In other words: \[a=1,\quad b=-1,\quad c=-6\] So: \[x^3-7x-6=(x+1)(x^2-x-6)\] By inspection we can factorise further: \[x^3-7x-6=(x+1)(x-3)(x+2)\] The roots of the cubic polynomial \(x^3-7x-6\) are therefore \(x=-1\), \(x=3\) and \(x=-2\).

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A second important property of a polynomial function concerns factorisation with real coefficients. For this purpose we first note that a polynomial \(p\) is irreducble if its degree is greater than 0 is and the polynomial cannot be written as a multiple of polynomials with lower nonzero degree, i.e., if the only divisors of \(p\) are constants and constant multiples of \(p\).

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A third important property of a polynomial function concerns asymptotic behaviour:

In the above example of \(f(x)=x^3-1\) is \(f(1000)=(1000)^3-1=999999999\approx 1\,000\,000\,000=1000^3\) .

More generally: For large \(x\), the sign of \(f(x)=a_nx^n+a_{n-1}x^{n-1} + \cdots + a_2x^2+a_1x+a_0\) is equal to the sign of the leading term \(a_nx^n\) and depends on

• the sign of \(x\) (positive or negative),
• the sign of \(a_n\), and
• oddness or evenness of \(n\) (After all, if \(x\) is a negative number, then \(x^n\) is negative if \(n\) is odd and positive for even \(n\)).