Which polynomial is of the fourth degree? 1) a^6+a^5b+a^4b^2+a^3b^3
2) a^3b+a^2b+b^3
3) a^4+a^2b^3
4) 4a^3

a^3b+a^2b+b^3  

Step-by-step explanation:

A fourth degree polynomial has the highest power of degree 4 when you sum the powers in each term

) a^6+a^5b+a^4b^2+a^3b^3   degree 6

2) a^3b+a^2b+b^3       3+1 = 3 degree 4

3) a^4+a^2b^3   2+3 = degree 5

4) 4a^3   degree 3

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!