Mathematics 10 — Exercise 5.2 · Q1 (viii)

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.

   $$2\times x^2-x-6\ne 0$$

2. Expand any powers or inner brackets before factoring.

   $$2\times x^2-x-6=2\times x^2-x-6$$

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

   $$(3-4=-1,3\times \left(-4\right)=-12)$$

4. Rewrite the middle term as those two terms.

   $$2\times x^2+3\times x-4\times x-6$$

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

   $$x\times \left(2\times x+3\right)-2\times \left(2\times x+3\right)$$

6. Factor out the common bracket.

   $$\left(x-2\right)\times \left(2\times x+3\right)$$

7. The complete factored form is:

   $$\left(x-2\right)\times \left(2\times x+3\right)$$

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

   $$3\times x^2-4\times x-4\ne 0$$

9. Expand any powers or inner brackets before factoring.

   $$3\times x^2-4\times x-4=3\times x^2-4\times x-4$$

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

    $$(2-6=-4,2\times \left(-6\right)=-12)$$

11. Rewrite the middle term as those two terms.

    $$3\times x^2+2\times x-6\times x-4$$

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

    $$x\times \left(3\times x+2\right)-2\times \left(3\times x+2\right)$$

13. Factor out the common bracket.

    $$\left(x-2\right)\times \left(3\times x+2\right)$$

14. The complete factored form is:

    $$\left(x-2\right)\times \left(3\times x+2\right)$$

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

    $$\left(x-2\right)\times \left(2\times x+3\right)\times \left(3\times x+2\right)$$

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

    $$\frac{6\times x^3+x^2-20\times x-12}{2\times x^2-x-6}=3\times x+2$$

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

    $$\frac{7\times \left(3\times x+2\right)}{\left(x-2\right)\times \left(2\times x+3\right)\times \left(3\times x+2\right)}$$

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

    $$\frac{6\times x^3+x^2-20\times x-12}{3\times x^2-4\times x-4}=2\times x+3$$

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

    $$\frac{-8\times \left(2\times x+3\right)}{\left(x-2\right)\times \left(2\times x+3\right)\times \left(3\times x+2\right)}$$

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

    $$\frac{7\times \left(3\times x+2\right)-8\times \left(2\times x+3\right)}{\left(x-2\right)\times \left(2\times x+3\right)\times \left(3\times x+2\right)}$$

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

    $$7\times \left(3\times x+2\right)=21\times x+14$$

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

    $$-8\times \left(2\times x+3\right)=-16\times x-24$$

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

    $$21\times x+14-(16\times x+24)=5\times x-10$$

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

    $$6\times x^3+x^2-20\times x-12\ne 0$$

25. Factor the numerator.

26. Expand any powers or inner brackets before factoring.

    $$5\times x-10=5\times x-10$$

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

    $$5\times x-10=5\times \left(x-2\right)$$

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

    $$5\times x-10=5\times \left(x-2\right)$$

29. The complete factored form is:

    $$5\times \left(x-2\right)$$

30. Factor the denominator.

31. Expand any powers or inner brackets before factoring.

    $$6\times x^3+x^2-20\times x-12=6\times x^3+x^2-20\times x-12$$

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

    $$6\times x^3+x^2-20\times x-12=\left(x-2\right)\times \left(2\times x+3\right)\times \left(3\times x+2\right)$$

33. The complete factored form is:

    $$\left(x-2\right)\times \left(2\times x+3\right)\times \left(3\times x+2\right)$$

34. Write the factored numerator over the factored denominator.

    $$\frac{5\times \left(x-2\right)}{\left(x-2\right)\times \left(2\times x+3\right)\times \left(3\times x+2\right)}$$

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

    $$x-2$$

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

    $$\frac{\frac{5\times x-10}{x-2}}{\frac{6\times x^3+x^2-20\times x-12}{x-2}}$$

37. Simplify the two divisions separately.

    $$\frac{5}{6\times x^2+13\times x+6}$$

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

    $$\frac{5}{\left(2\times x+3\right)\times \left(3\times x+2\right)}$$

Answer

The final answer is shown below.