Examples from Complex polynomials

Classify and find degrees

The expressions

are complex polynomials. The non-zero polynomials $z$, $z^{999}$, $1+35z^4$ and $i+\sqrt2\,iz^4+\pi z$ have degrees $1$, $999$, $4$ and $4$, respectively. The zero polynomial has undefined degree.

An expression containing a fractional power such as $z^{1/2}$ is not a polynomial in $z$. By contrast, numbers such as $2+3i$ and $(1-7i)^{17}$ are allowed as coefficients: only the powers of the variable must be non-negative integers.

Find and use roots

For $p(z)=z^2+1$,

so $i$ is a root. Direct substitution also gives $p(-i)=0$. The factor theorem below explains why these roots yield the factorisation

into linear factors. The same polynomial has no real root: for real $x$, $x^2+1>0$.

Multiplicity two

For $p(z)=(z-1)^2$, the root $1$ has multiplicity two: $(z-1)^2$ is a factor of $p$ but $(z-1)^3$ is not.

Multiplicity records repetition in a factorisation. It is not the number of different solutions: this polynomial has one distinct root, counted twice when roots are counted with multiplicity.

Factorise a quadratic

For $p(z)=z^2+z+1$, the quadratic formula gives

Therefore

Factorise $z^8-1$

The roots satisfy $z^8=1$, so the eighth roots of unity give

Hence

over $\mathbb C$.

Factorise $x^8-1$ over the reals

The real roots are $1$ and $-1$. Pairing the remaining eighth roots with their conjugates gives

The complex factorisation and the real factorisation describe the same polynomial; the latter groups conjugate roots into quadratic factors.