How do you simplify #\frac { x ^ { 2} - 3x + 1} { x + 2} \cdot \frac { 2x - 1} { x ^ { 2} - 3}#?
1 Answer
#(x^2-3x+1)/(x+2) * (2x-1)/(x^2-3)=2-(11(x^2-x-1))/(x^3+2x^2-3x-6)#
Explanation:
Given:
#(x^2-3x+1)/(x+2) * (2x-1)/(x^2-3)#
Normally, when given a rational expression to simplify you would factor the polynomials, identify common factors and cancel them.
The given rational expression has no common factors, as we can see by identifying all of the zeros of the numerators and denominators:
-
#x^2-3x+1 = (x-3/2)^2-(sqrt(5)/2)^2 = (x-3/2-sqrt(5)/2)(x-3/2+sqrt(5)/2)# has zeros#x=3/2+-sqrt(5)/2# -
#2x-1# has zero#x=1/2# -
#x+2# has zero#x=-2# -
#x^2-3# has zeros#x=+-sqrt(3)#
With no common factors, we can simplify a different way:
Multiply out all of the factors, then divide to give a quotient and remainder.
#(x^2-3x+1)/(x+2) * (2x-1)/(x^2-3)#
#=(2x^3-7x^2+5x-1)/(x^3+2x^2-3x-6)#
#=(2x^3+4x^2-6x-12-11x^2+11x+11)/(x^3+2x^2-3x-6)#
#=(2(x^3+2x^2-3x-6)-11(x^2-x-1))/(x^3+2x^2-3x-6)#
#=2-(11(x^2-x-1))/(x^3+2x^2-3x-6)#
