1/04/2023

Polynomial and Rational Functions

 A gold bar maker wants to design a 14-kg gold bar for its customers. Assume the density of gold is 1 kg/m³, the height of the bar is x, the length is designed to be 3x+4, and the width is x+1, how do we know the height of the bar should be?

The weight of the bar equals the volume of the bar times the density of gold. The volume of the bar equals is the product of its height, length, and its width. Therefore, we have....

  x(3x+4)(x+1)1=14

  x(3x^2 +7x +4)=14

 3x^3 +7x^2 +4x -14=0

The factor of  3x^3  is   \pm1 \pm3

The factor of  -14  is   \pm1 \pm2 \pm7

Test these factors start with 1...

 3(1)^3 +7(1)^2 +4(1) -14=0  ,  fortunately, we have a real zero at x=1

 3x^3 +7x^2 +4x -14=(x-1)(3x^2 +10x +14)=0



On the graph, there is only one x-intercept at x=1
However, the quotient  3x^2 +10x +14  does not not have any intercept at y=0


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