Sort the following polynomials, specifie their degree and find out if they are finished and/or homogeneous:
- $$p(x)=3x-3x^2+1+x^4$$
- $$q(x,y)=3xy-3x^2-4y^2$$
- $$r(x,y)=10-x-xy-x^2-\dfrac{2}{5}y^2+y$$
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.