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?

$$x^6+10$$ $$x+1$$

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.

$$2x^2+x-1$$ that is the result of adding the monomials $2x^2$ and $x$, and subtracting the monomial $1$.

Or $$3x^5-x^2+x-5$$ that is the result of adding the monomials $3x^5$ and $x$, and subtracting the monomials $x^2$ and $5$.

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:

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

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

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

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

$q(x,y)=3x^2y+4x$, $q(x)=3x^2y+4x$

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. $$q(x)=-1$$ $$q(x)=\dfrac{1}{2}$$
• The first degree: $$q(x)=x-1$$ $$q(y)=3y-\dfrac{3}{4}$$ $$p(y)=\dfrac{y}{2}+\dfrac{1}{4}$$
• The second degree: $$p(z)=z^2+3z-9$$ $$p(x)=\dfrac{x^2}{3}+2x$$ $$q(z)=z^2-\dfrac{10}{3}$$
• Third degree: $$r(t)=t^3+t^2+1$$ $$p(t)=\dfrac{t^3}{4}+\dfrac{t^2}{2}-t+10$$ $$q(x)=x^3-\dfrac{1}{4}$$

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. $$p(x)=x^3+x^2+x+1$$ $$p(x,y)=2x^2+y^2-xy+x+y-\dfrac{1}{3}$$ $$r(t)=t^2-4t+9$$
• Incomplete polynomial: it has some coefficient which value is zero. $$p(x)=x^3+x+1$$ $$p(x,y)=2x^2+y^2+x+y-\dfrac{1}{3}$$ $$r(t)=t^2-4t$$
• 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. $$p(x)=2x$$ $$p(x,y)=3x^2y+4x^3+2xy^2$$ $$p(x,y)=\dfrac{xy}{2}+x^2+y^2$$
• Heterogeneous polynomial: not all their monomials have the same degree. $$p(x)=2x-1$$ $$p(x,y)=3x^2y+4x^3+2xy^2$$ $$p(x,y)=\dfrac{xy}{2}+x^2y+y^2$$
• 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. $$p(x)=3x^2+1$$ $$q(x)=1+3x^2$$ $$p(x,y)=xy+4x-1$$ $$q(y,x)=-1+4x+yx$$