The flavor polynom(coeffs, x) describes a random polynomial in a given number of variables x with coefficients coeffs of a given random flavor.

To describe a flavor of a random polynomial, use either polynom(coeffs, x) or polynom(coeffs, x, opt) (where opt is described following) as the argument to RandomTools[Generate] or as part of a structured flavor.

By default, the flavor polynom(coeffs, x) describes a random polynomial in the given variable(s) of degree $5$.

You can modify the properties of the random polynomial by specifying opt as the equation $\mathrm{degree}=n$, where n is a non-negative integer that indicates the degree of the polynomial. By default, the degree of the polynomial is set to $5$.

$\mathrm{with}\left(\mathrm{RandomTools}\right)\:$

$\mathrm{Generate}\left(\mathrm{polynom}\left(\mathrm{rational}\,x\right)\right)$

$-\frac{104281139459}{499999999994}-\frac{153430091789}{249999999997}x-\frac{238787914764}{249999999997}{x}^2+\frac{150093742233}{249999999997}{x}^3-\frac{36223971562}{249999999997}{x}^4+\frac{342622684449}{499999999994}{x}^5$

$\mathrm{Generate}\left(\mathrm{polynom}\left(\mathrm{positive}\left(\mathrm{denominator}=12\right)\,x\,\mathrm{degree}=10\right)\right)$

$\frac{1}{12}+\frac{2}{3}x+\frac{5}{12}{x}^2+\frac{5}{6}{x}^3+\frac{11}{12}{x}^4+\frac{1}{6}{x}^5+\frac{1}{6}{x}^6+\frac{1}{3}{x}^7+\frac{2}{3}{x}^8+\frac{11}{12}{x}^9+\frac{1}{4}{x}^{10}$

$\mathrm{Generate}\left(\mathrm{polynom}\left(\mathrm{nonnegint}\left(\mathrm{range}=2\right)\,\left\{x\,y\right\}\right)\right)$

$y{x}^4+2y^2{x}^3+2x^2{y}^3+2x{y}^4+2y^5+y{x}^3+2x^2{y}^2+y^4+x^3+y{x}^2+2x{y}^2+x^2+xy+2y^2+x$