Mathematics 10 — Exercise 5.2 · Q1 (ii)

Full question

Simplify the following sum or difference of algebraic fractions.

Step-by-step solution

1. Fraction 1: keep its denominator nonzero and factor that denominator.

   $$x^2-3\times x+2\ne 0$$

2. Expand any powers or inner brackets before factoring.

   $$x^2-3\times x+2=x^2-3\times x+2$$

3. Split the middle coefficient: these two numbers add to b and multiply to a·c.

   $$(-1-2=-3,\left(-1\right)\times \left(-2\right)=2)$$

4. Rewrite the middle term as those two terms.

   $$x^2-x-2\times x+2$$

5. Group two terms at a time and take each common factor.

   $$x\times \left(x-1\right)-2\times \left(x-1\right)$$

6. Factor out the common bracket.

   $$\left(x-2\right)\times \left(x-1\right)$$

7. The complete factored form is:

   $$\left(x-2\right)\times \left(x-1\right)$$

8. Fraction 2: keep its denominator nonzero and factor that denominator.

   $$1-x\ne 0$$

9. Fraction 3: keep its denominator nonzero and factor that denominator.

   $$x-2\ne 0$$

10. Build the least common denominator from every distinct denominator factor at its largest required power.

    $$\left(x-2\right)\times \left(x-1\right)$$

11. For fraction 1, divide the common denominator by its own denominator.

    $$\frac{x^2-3\times x+2}{x^2-3\times x+2}=1$$

12. Multiply this numerator by the same factor to keep the fraction unchanged.

    $$\frac{1\times \left(4\times x\right)}{\left(x-2\right)\times \left(x-1\right)}$$

13. For fraction 2, divide the common denominator by its own denominator.

    $$\frac{x^2-3\times x+2}{1-x}=2-x$$

14. Multiply this numerator by the same factor to keep the fraction unchanged.

    $$\frac{-4\times \left(2-x\right)}{\left(x-2\right)\times \left(x-1\right)}$$

15. For fraction 3, divide the common denominator by its own denominator.

    $$\frac{x^2-3\times x+2}{x-2}=x-1$$

16. Multiply this numerator by the same factor to keep the fraction unchanged.

    $$\frac{-5\times \left(x-1\right)}{\left(x-2\right)\times \left(x-1\right)}$$

17. The denominators now match. Put all signed numerators over one common denominator.

    $$\frac{1\times \left(4\times x\right)-4\times \left(2-x\right)-5\times \left(x-1\right)}{\left(x-2\right)\times \left(x-1\right)}$$

18. Expand numerator product 1, keeping minus signs on the whole bracket.

    $$1\times \left(4\times x\right)=4\times x$$

19. Expand numerator product 2, keeping minus signs on the whole bracket.

    $$-4\times \left(2-x\right)=4\times x-8$$

20. Expand numerator product 3, keeping minus signs on the whole bracket.

    $$-5\times \left(x-1\right)=5-5\times x$$

21. Collect like powers and constants in the combined numerator.

    $$4\times x+4\times x-8-(5\times x-5)=3\times x-3$$

22. The original denominator must be nonzero. Keep this restriction even if a factor is later cancelled.

    $$x^2-3\times x+2\ne 0$$

23. Factor the numerator.

24. Expand any powers or inner brackets before factoring.

    $$3\times x-3=3\times x-3$$

25. Take the common factor out of every term. Show each remaining term after division.

    $$3\times x-3=3\times \left(x-1\right)$$

26. Apply the matching factor identity: difference of squares, sum/difference of cubes, or grouping. Expand the displayed factors to check the identity.

    $$3\times x-3=3\times \left(x-1\right)$$

27. The complete factored form is:

    $$3\times \left(x-1\right)$$

28. Factor the denominator.

29. Expand any powers or inner brackets before factoring.

    $$x^2-3\times x+2=x^2-3\times x+2$$

30. Split the middle coefficient: these two numbers add to b and multiply to a·c.

    $$(-1-2=-3,\left(-1\right)\times \left(-2\right)=2)$$

31. Rewrite the middle term as those two terms.

    $$x^2-x-2\times x+2$$

32. Group two terms at a time and take each common factor.

    $$x\times \left(x-1\right)-2\times \left(x-1\right)$$

33. Factor out the common bracket.

    $$\left(x-2\right)\times \left(x-1\right)$$

34. The complete factored form is:

    $$\left(x-2\right)\times \left(x-1\right)$$

35. Write the factored numerator over the factored denominator.

    $$\frac{3\times \left(x-1\right)}{\left(x-2\right)\times \left(x-1\right)}$$

36. This is the common factor of numerator and denominator.

    $$x-1$$

37. Divide numerator and denominator by that same nonzero factor.

    $$\frac{\frac{3\times x-3}{x-1}}{\frac{x^2-3\times x+2}{x-1}}$$

38. Simplify the two divisions separately.

    $$\frac{3}{x-2}$$

39. Write the fraction in lowest form. Only factors, never added terms, have been cancelled.

    $$\frac{3}{x-2}$$

Answer

The final answer is shown below.