The area of a rectangle is #(x^4+4x^3+ 3x^2-4x-4)#, and the length of the rectangle is #(x^3 +5x^2 +8x+ 4)#. If area length x width, what is the width of the rectangle?

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Answer 1 · 2016-12-28T09:52:39

#x-1#

The width is given by the rational expression:

#(x^4+4x^3+3x^2-4x-4)/(x^3+5x^2+8x+4)#

Let us see if we can simplify the expression:

Note that #x=1# is a zero of the numerator, since the sum of the coefficients is zero. That is:

#1+4+3-4-4 = 0#

So #(x-1)# is a factor:

#x^4+4x^3+3x^2-4x+4 = (x-1)(x^3+5x^2+8x+4)#

So:

#(x^4+4x^3+3x^2-4x-4)/(x^3+5x^2+8x+4) = x-1#