# 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. $$p(x)=3x^2+1$$$ $$q(x)=1+3x^2$$$$$p(x,y)=xy+4x-1$$$ $$q(y,x)=-1+4x+yx$$\$