IB Maths AA 2.5 Notes

Polynomial functions

Advanced functions build on earlier work with graphs, roots and equations. The main focus here is polynomial functions, the factor and remainder theorems, sums and products of roots, further rational functions, and odd and even functions.

A polynomial function is a function of the form:

$f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$

where $n$ is a non-negative integer and the coefficients are real numbers.

Examples include:

• $f(x)=x^2-5x+6$
• $g(x)=2x^3-x+4$
• $h(x)=x^4-3x^2+1$

The highest power of $x$ is called the degree of the polynomial.

A polynomial may have roots, also called zeros, where $f(x)=0$. These are the $x$-intercepts of the graph. If $x=a$ is a root, then $(x-a)$ is a factor of the polynomial.

These ideas are closely linked:

• If $f(a)=0$, then $a$ is a root or zero.
• If $a$ is a root, then $(x-a)$ is a factor.
• If $(x-a)$ is a factor, then $f(a)=0$.

Find the roots and factors of $f(x)=x^2-5x+6$.

Factorise: $f(x)=(x-2)(x-3)$.

So the roots are $x=2$ and $x=3$.

The corresponding factors are $(x-2)$ and $(x-3)$.