# Definition and classification of polynomials

## Problems

Sort the following polynomials, specifie their degree and find out if they are finished and/or homogeneous:

1. $p(x)=3x-3x^2+1+x^4$
2. $q(x,y)=3xy-3x^2-4y^2$
3. $r(x,y)=10-x-xy-x^2-\dfrac{2}{5}y^2+y$
See development and solution

### Development:

1) 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.

2) 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 .

3) 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 .

### Solution:

1) 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.

2) Ordered: Yes

Degree: $2$

Completed: No. Elements of degree $1$ and $0$ are missing.

Homogeneous: Yes.

3) 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.

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