Mathematics 10 — Exercise 5.2 · Q1 (v)

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.

   $$x+1\ne 0$$

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

   $$x+2\ne 0$$

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

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

4. Expand any powers or inner brackets before factoring.

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

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

   $$(2+1=3,2\times 1=2)$$

6. Rewrite the middle term as those two terms.

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

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

   $$x\times \left(x+2\right)+x+2$$

8. Factor out the common bracket.

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

9. The complete factored form is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    $$x+2=x+2$$

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

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

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

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

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

    $$x+2-(2\times x+2)-(2\times x+3)=-3\times x-3$$

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.

    $$(2+1=3,2\times 1=2)$$

31. Rewrite the middle term as those two terms.

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

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

    $$x\times \left(x+2\right)+x+2$$

33. Factor out the common bracket.

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

34. The complete factored form is:

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

35. Write the factored numerator over the factored denominator.

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

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.