Prism is a polyhedron that has two parallel polygonal faces. The area of the base can be written as (x-2)(2x+9).
A prism is a polyhedron that has two polygonal faces lying in parallel planes while the other faces are parallelograms.
As it is given that the volume of the prism is 2x³ + 9x² -8x -36 while its height is x+2, therefore, in order to calculate the area of the base we need to divide the polynomial with the height of the prism.
As we need to divide the polynomial 2x³ + 9x² -8x -36 with x+2, therefore, Using the synthetic division the polynomial is divided as shown below.
The polynomial can be factorized as,
[tex]2x^3+9x^2-8x-36\\\\x^2(2x+9)-4(2x+9)\\\\(x^2-4)(2x+9)\\\\(x+2)(x-2)(2x+9)[/tex]
Thus, the division of the polynomial can be written as,
[tex]\dfrac{2x^3+9x^2-8x-36}{x+2} = \dfrac{(x+2)(x-2)(2x+9)}{(x+2)}=(x-2)(2x+9)[/tex]
Hence, the area of the base can be written as (x-2)(2x+9).
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