Use the graph of the polynomial function to find the factored form of
its related polynomial. Assume it
has no constant factor

The factored form of the polynomial is .

Step-by-step explanation:

According to the graph, we have a second order polynomial with a vertical axis of symmetry. The standard form of the polynomial is:

(1)

Where:

- Independent variable.

- Dependent variable.

, , - Coefficients.

We can obtain the solution of the polynomial by knowing three distinct points. From graph we know that the curve pass through the following three points: , , . Three linear equations are constructed:

(2)

(3)

(4)

The solution of this system is , and . Then, the polynomial in standard form is . By factorization, we find that factored form is:

(5)

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!