The length of the rectangular delivery box must be equal to 4 inches.
Let the length of the box be L.
Let the width of the box be W.
Let the height of the box be H.
Given the following data:
Translating the word problem into an algebraic expression;
[tex]W = 3 + L[/tex] ......equation 1
[tex]H = 4 + L[/tex] ......equation 2.
Mathematically, the volume of a rectangular solid is given by the formula;
[tex]Volume = length * width * height[/tex] .....equation 3.
Substituting the values into equation, we have;
[tex]224 = L * (3 + L) * (4 + L)\\\\224 = (3L + L^{2})* (4 + L)\\\\224 = 12L + 3L^{2} + 4L^{2} + L^{3} \\\\224 = 12L + 7L^{2} + L^{3}[/tex]
Rearranging the polynomial, we have;
[tex]L^{3} + 7L^{2} + 12L - 224 = 0[/tex]
We would apply the remainder theorem to solve the polynomial.
According to the remainder theorem, if a polynomial P(x) is divided by (x - r) and there is a remainder R; then P(r) = R.
When x = 3
[tex](x - 3) = 0\\x = 3[/tex]
[tex]P(3) = 3^{3} + 7(3^{2}) + 12(3) - 224\\\\P(3) = 27 + 7(9) + 36 - 224\\\\P(3) = 27 + 63 + 36 - 224 = -98 \neq 0[/tex]
We would try with 4;
[tex]P(4) = 4^{3} + 7(4^{2}) + 12(4) - 224\\\\P(4) = 64 + 7(16) + 48 - 224\\\\P(4) = 64 + 112 + 48 - 224\\\\P(4) = 224 - 224 = 0[/tex]
Therefore, 4 is one of its roots.
Hence, the length of the rectangular delivery box must be equal to 4 inches.
Find more information on polynomial here: https://brainly.com/question/10689855
