Fundamental Theorem of Algebra (FTA)

(Writing Polynomials in Root-Factored Form)

We now learn about the Fundamental Theorem of Algebra (FTA), which, alongside its corollary and the factor theorem, will allow us to solve any polynomial equation as well as root factor any polynomial.

Another, and perhaps simpler, way of thinking of the fundamental theorem of algebra (FTA) is that when working within \(\mathbb{C}\) (the set of complex numbers) an equation of the form: \[a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x+a_0 = 0\] will always have at least one solution.

Careful, if we're "only" working within \(\mathbb{R}\) (the set of real numbers) the FTA is no longer true. Indeed, if we're working within the set or real numbers, there is no rule in \(\mathbb{R}\) that says that a polynomial equation must have a solution. An example of this is quickly shown with the following equation: \[x^2+4 = 0\] in \(\mathbb{R}\) this equation has no solution, which illustrates the fact that in \(\mathbb{R}\) the FTA isn't true. On the other hand, if we're working in \(\mathbb{C}\) it has two solutions: \(x = -2i\) and \(x = 2i\).

Root Factoring Polynomials

As we've just seen, the polynomial \(f(x)\) can be written \(f(x) = \begin{pmatrix}x - c_1 \end{pmatrix}.Q_1(x)\). But since \(Q_1(x)\) is itself a polynomial function we can apply the FTA to it and state that there is a value \(x\), say \(x = c_2\), such that: \[Q_1(c_2)=0\] and once more we can use the corrolary of the FTA, this time to state that: \[Q_1(x) = \begin{pmatrix}x - c_2 \end{pmatrix}.Q_2(x)\] and therefore, going back to \(f(x)\), we can write: \[f(x) = \begin{pmatrix}x - c_1 \end{pmatrix}.Q_1(x) = \begin{pmatrix}x - c_1 \end{pmatrix}.\begin{pmatrix}x - c_1\end{pmatrix}.Q_2(x) \] Following the same "pattern" \(n\) times, for \(Q_2(x)\), \(Q_3(x)\), ... \(Q_n(x)\) leads to writing the polynomial function in its root factored form: \[f(x) = a_n\begin{pmatrix}x - c_1\end{pmatrix}.\begin{pmatrix}x - c_2\end{pmatrix}. \dots . \begin{pmatrix}x - c_n\end{pmatrix}\] Where : \[a_n = Q_n(x)\] The idea behind root factoring polynomials is written more formally in corollary 2.

What this means

This corallary is it is telling us that a polynomial of degree \(n\) can be written as a product of \(n\) linear factors. This also means that the polynomial has \(n\) zeros (roots).

Careful: this is true so long as we are accepting roots to be complex numbers, in \(\mathbb{C}\). If we're working exclusively with real numbers then the fundamental theorem of algebra tells us that a polynomial function of degree \(n\): \[f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x+a_0\] can have upto \(n\) real roots.

Example 1

The polynomial function: \[f(x)= 3x^3 - 6x^2 - 3x + 6\] Has \(3\) real roots: \[\left \{-1, \ 1, \ 2 \right \}\] In other words: \[f(-1)=f(1)=f(2) = 0\] And we can write this polynomial in its root factored form as: \[f(x) = 3.\begin{pmatrix}x+1\end{pmatrix}.\begin{pmatrix}x-1\end{pmatrix}.\begin{pmatrix}x-2\end{pmatrix}\]

Example 2

The polynomial function: \[f(x) = 2x^5 - 12x^4 + 30x^3 - 60x^2 + 88x - 48\] has \(3\) real roots and \(2\) complex roots: \[\left \{1, \ 2, \ 3, \ -2i, \ 2i \right \}\] in other words: \[f(1)=f(2)=f(3) = f(-2i) = f(2i)= 0\] In its root factored form, this polynomial can therefore be written: \[f(x) = 2.\begin{pmatrix}x-1\end{pmatrix}.\begin{pmatrix}x-2\end{pmatrix}.\begin{pmatrix}x-3\end{pmatrix}.\begin{pmatrix}x-2i\end{pmatrix}.\begin{pmatrix}x+2i\end{pmatrix}\]

In both of the examples, above, the polynomials has distinct roots; the cubic has \(3\) distinct roots and the quintic polynomial had \(5\) distinct roots. That won't always be the case.

Example

For instance the cubic polynomial: \[f(x) = x^3 - 3x^2 + 3x - 1\] has three identical roots at \(x = 1\). In root-factored form we can write: \[f(x) = \begin{pmatrix}x - 1\end{pmatrix}.\begin{pmatrix}x - 1\end{pmatrix}.\begin{pmatrix}x - 1\end{pmatrix}\] Or "simply": \[f(x) = \begin{pmatrix}x - 1\end{pmatrix}^3\] We say that the root has multiplicity of \(3\).