Mathematics
shaylynnharcrow
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The volume of a rectangular prism is b3 + 8b2 + 19b + 12 cubic units, and its height is b + 3 units. The area of the base of the rectangular prism is square units.

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GalaxyVanz

Given: The volume of the rectangular prism is  [latex] b^{3}+8 b^{2} +19b+12 [/latex], the height is h=(b+3) 1. The volume of a rectangular prism is (base area)*height also, notice that the volume is a third degree polynomial, the height is a 1st degree polynomial, so the base area must be a 2nd degree polynomial, whose coefficients we don't know yet. Let this quadratic polynomial be [latex](mb^{2}+nb+k)[/latex]   2 [latex]b^{3}+8 b^{2} +19b+12=(mb^{2}+nb+k)*(b+3)[/latex] notice that  [latex]b^{3}[/latex] is the product of the largest 2 terms: [latex]mb^{2}[/latex] and b, so m must be 1 also, notice that 12 is the product of the constants, k and 3 so k*3=12, this means k=4 3 we write the above equality again: [latex] b^{3}+8 b^{2} +19b+12=(b^{2}+nb+4)*(b+3)[/latex] [latex](b^{2}+nb+4)(b+3)= b^{3}+3 b^{2} +nb^{2}+3nb+4b+12[/latex] =[latex]= b^{3}+(n+3)b^{2}+(3n+4)b+12[/latex] 4 now compare the coefficient with the left side: [latex]8 b^{2}=(n+3)b^{2}[/latex] 8=n+3 n=5 substituting n=5:  the base area is [latex]b^{2}+5b+4 [/latex] Answer: [latex]b^{2}+5b+4 [/latex]

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