Part a.) the area of a square is (16x^2-8x+1) square units. determine the length of each side of the square by factoring the areas expressions completely. show all work

part b.) the area of a rectangle is (81x^2-4y^2) square units. determine the dimensions of the rectangle by factoring the area of the expression. show all work

part c.) the volume of a rectangular box is (x^3-9x^2-4x+36) cubic units. determine the dimensions of the rectangular box by factoring the volume expression completely. show all work

Answers are

a) the area of a square is A = L^2 = 16x^2 - 8x+1 = (4x - 1) ^2, so sqrt (L^2) = sqrt (4x - 1) ^2, and so L = 4x - 1,

b) the area of a rectangle is A = h x w = 81x^2-4y^2 = (9x - 2y) (9x + 2y)

so w = (9x - 2y) and h = (9x + 2y)

c) let be f (x) = x^3-9x^2-4x+36, so f (2) = 8-36-8+36=0, f (x) can be written as

f (x) = (x - 2) (ax^2 + bx + c), we must find a, b and c

(ax^2 + bx + c) = x^3-9x^2-4x+36 / x - 2, after the division we have

ax^2 + bx + c = x^2-7x-18, and x = - 2 is a zero of x^2-7x-18, so x^2-7x-18 = (x + 2) (ax + b), ax + b = x^2-7x-18 / x + 2 = x - 9

finally V = h x w x L = (x - 2) (x + 2) (x - 9),