Definition and classification of polynomials

When we multiply a number (coefficient) for an unknown (variable) is a monomial. But what if we add instead of multiply?

What happens when we add monomials that are similar? and if we subtract them?

When we join not similar monomials by adding or subtracting them we get a polynomial.

In mathematics, to call polynomials we use one letter followed by a parenthesis with the variable (or variables, separated by commas). So the above examples would be:

$p(x)=2x^2+x-1$ and $q(x)=3x^5-x^2+x-5$

If there is more than one variable, as we said:

$p(x,y)=x^{6}y+xy-x$

$q(x,y,z)=xy{z}^2+xyz-x{y}^{3}z-zyz+zy-z$

$r(x,y,z,t)=xyzt$

Be careful in the way we represent polynomials because it is easy to make notation mistakes.

In the first polynomial, "$y$" acts as a variable. However, in the second, the "$y$" is a coefficient (which value is $y$, a number that we don't know a priori).

So they are two different polynomials (For example, the first one has degree $3$ and the second one has degree $2$).

Now, using as an example the polynomial $p(x)=2x^2+x-1$, we define the following characteristics of a polynomial:

• Variable/s of the polynomial: unknown or unknowns that we find in the polynomial. In the polynomial $p(x),x$.

• Degree of the polynomial: the greatest exponent of all monomials which has the polynomial. In our example $max\{2,1,0\}=2$

• Leading coefficient: the coefficient of the monomial which has the higher degree. In our case, $2$.

• Independent term: the coefficient of the monomial with exponent zero. If there is no such monomial then is equal to $0$. In our case, it is $-1$.

Classification of polynomials

We can categorize the polynomials according to their characteristics.

Classification of polynomials according to their degree

• Degree zero: They are coefficients. $$q(x)=-1$$ $$q(x)={\displaystyle \frac{1}{2}}$$
• The first degree: $$q(x)=x-1$$ $$q(y)=3y-{\displaystyle \frac{3}{4}}$$ $$p(y)={\displaystyle \frac{y}{2}}+{\displaystyle \frac{1}{4}}$$
• The second degree: $$p(z)=z^2+3z-9$$ $$p(x)={\displaystyle \frac{x^2}{3}}+2x$$ $$q(z)=z^2-{\displaystyle \frac{10}{3}}$$
• Third degree: $$r(t)=t^3+t^2+1$$ $$p(t)={\displaystyle \frac{t^3}{4}}+{\displaystyle \frac{t^2}{2}}-t+10$$ $$q(x)=x^3-{\displaystyle \frac{1}{4}}$$

And, in this way, we might continue to the number that we need.

Classification of polynomials according to their coefficients

• Finished polynomial: it has all the coefficients other than zero. $$p(x)=x^3+x^2+x+1$$ $$p(x,y)=2x^2+y^2-xy+x+y-{\displaystyle \frac{1}{3}}$$ $$r(t)=t^2-4t+9$$
• Incomplete polynomial: it has some coefficient which value is zero. $$p(x)=x^3+x+1$$ $$p(x,y)=2x^2+y^2+x+y-{\displaystyle \frac{1}{3}}$$ $$r(t)=t^2-4t$$
• Null polynomial: all its coefficient are equal to zero. $$p(x)=0$$

Classification of polynomials according to the degrees of their monomials

• Ordered polynomial: the monomials are written from greater to lesser degree. $$p(x)=x^4+x^3+x^2+x+1$$ $$q(x)=x^6+x^4+x^2+x+1$$ $$r(x)=x^{100}+x^2+2x$$
• Homogeneous polynomial: all their monomials have the same degree. $$p(x)=2x$$ $$p(x,y)=3x^{2}y+4x^3+2x{y}^2$$ $$p(x,y)={\displaystyle \frac{xy}{2}}+x^2+y^2$$
• Heterogeneous polynomial: not all their monomials have the same degree. $$p(x)=2x-1$$ $$p(x,y)=3x^{2}y+4x^3+2x{y}^2$$ $$p(x,y)={\displaystyle \frac{xy}{2}}+x^{2}y+y^2$$
• Equal polynomials. They are those which satisfy the next conditions:
  • They have the same degree.
  • The coefficients of the monomials of the same degree are equal. $$p(x)=3x^2+1$$ $$q(x)=1+3x^2$$ $$p(x,y)=xy+4x-1$$ $$q(y,x)=-1+4x+yx$$