Definition and classification of polynomials
Sort the following polynomials, specifie their degree and find out if they are finished and/or homogeneous:
- $p(x)=3x-3x^2+1+x^4$
- $q(x,y)=3xy-3x^2-4y^2$
- $r(x,y)=10-x-xy-x^2-\dfrac{2}{5}y^2+y$
- Ordered: No, an element of degree 1 $(3x)$ is placed before an element of degree 2 $(-3x^2)$.
Degree: $max\{1,2,0,4\}$.
Completed: Degree elements $1,2,0,4$ exist.
Homogeneous: We can find different degree elements.
- Ordered: Yes, every element has degree $2$.
Degree: $max\{2,2,2\}$.
Completed: We can only find elements of degree $2$.
Homogeneous: All the elements have the same degree .
- Ordered: No, there is an element of degree zero $(10)$ that is placed before an element of degree one $(-x)$.
Degree: $max\{0,1,2,2,2,1\}$.
Completed: There are elements of degree $1,2,0$.
Homogeneous: There are elements of different degree .
- The ordered polynomial would be $p(x)=x^4-3x^2+3x+1$
Degree: $4$
Completed: No. An element of degree $3$ is missing.
Homogeneous: No.
- Ordered: Yes
Degree: $2$
Completed: No. Elements of degree $1$ and $0$ are missing.
Homogeneous: Yes.
- The ordered polynomial would be $r(x,y)=-xy-x^2-\dfrac{2}{5}y^2-x+y+10$
Degree: $2$
Completed: Yes.
Homogeneous: No.