Simplify the function
f(x)=(x−1)−(x−2)(x−1)(x−2)+(x−2)−(x−3)(x−2)(x−3)+(x−3)−(x−4)(x−3)(x−4)+(x−4)−(x−5)(x−4)(x−5).
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. (x−1)−(x−2)(x−1)(x−2). 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(x−2)(x−3)
7(x−1)(x−8)
- 1(x−2)(x−8)
How does the fraction (x−2)−(x−3)(x−2)(x−3) help?
Do you think all these fractions can be written in the form A(x−p)+B(x−q)?