Show that (2x^2 + x -15)/(2x^3 +6x^2) * 6x^3/(2x^2 - 11x + 15) simplifies to ax/(x + b) where a and b are integers
As you begin the question, it is important to note that immediately multiplying the two fractions, while would eventually get you to the correct answer would take far too long in an exam scenario. Instead, what should be done is beginning by factorising the two fractions. This gets you to (2x - 5)(x + 3)/2x2(x+3) * 6x3/(2x - 5)(x - 3). From this, you can see a common multiple of x + 3 in the first fraction and so it can be cancelled to give you (2x - 5)/2x2 * 6x3/(2x -5)(x - 3). These fractions are now each in their own individual simplest form and so can be multiplied together but not expanding the brackets. This then gives a single fraction of 6x3(2x - 5)/2x2(2x - 5)(x - 3). In this there is another common multiple of 2x - 5 and so this can be cancelled to give you 6x3/2x2(x - 3). Finally, 2x2 is a common multiple in the numerator and denominator and so the fraction can be simplified into its final form of 3x/(x - 3) to which there are no common multiples between. This means that the value of a is 3 and b is -3.
