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Simplify the function

f(x)=(x1)(x2)(x1)(x2)+(x2)(x3)(x2)(x3)+(x3)(x4)(x3)(x4)+(x4)(x5)(x4)(x5).

Think about the following questions. Some may be more helpful than others.

  • Can a common denominator be found?

  • What happens if we simplify the numerator? What might the next step be?

  • Look at the individual fractions, e.g. (x1)(x2)(x1)(x2). Can it be split into simpler fractions?

Is it possible to split each of the following fractions up into the sum of two fractions?

  • 1(x2)(x3)

  • 7(x1)(x8)

  • 1(x2)(x8)

How does the fraction (x2)(x3)(x2)(x3) help?

Do you think all these fractions can be written in the form A(xp)+B(xq)?