Mathematics 10 — Exercise 5.3 · Q1 (vii)

Full question

Simplify the following product or quotient of algebraic fractions, retaining the restrictions on the original denominators and divisors.

Step-by-step solution

1. Fraction 1: its original denominator cannot be zero.

   $$x^2-x-20\ne 0$$

2. Expand any powers or inner brackets before factoring.

   $$x^2-x-6=x^2-x-6$$

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

   $$(2-3=-1,2\times \left(-3\right)=-6)$$

4. Rewrite the middle term as those two terms.

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

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

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

6. Factor out the common bracket.

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

7. The complete factored form is:

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

8. Expand any powers or inner brackets before factoring.

   $$x^2-x-20=x^2-x-20$$

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

   $$(4-5=-1,4\times \left(-5\right)=-20)$$

10. Rewrite the middle term as those two terms.

    $$x^2+4\times x-5\times x-20$$

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

    $$x\times \left(x+4\right)-5\times \left(x+4\right)$$

12. Factor out the common bracket.

    $$\left(x-5\right)\times \left(x+4\right)$$

13. The complete factored form is:

    $$\left(x-5\right)\times \left(x+4\right)$$

14. Fraction 2: its original denominator cannot be zero.

    $$x^2+4\times x\ne 0$$

15. A divisor fraction must also be nonzero, so its numerator cannot be zero.

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

16. Change division by this fraction into multiplication by its reciprocal.

    $$\frac{x^2+4\times x}{x^3-3\times x^2}$$

17. Expand any powers or inner brackets before factoring.

    $$x^2+4\times x=x^2+4\times x$$

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

    $$x^2+4\times x=x\times \left(x+4\right)$$

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

    $$x^2+4\times x=x\times \left(x+4\right)$$

20. The complete factored form is:

    $$x\times \left(x+4\right)$$

21. Expand any powers or inner brackets before factoring.

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

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

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

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

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

24. The complete factored form is:

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

25. Fraction 3: its original denominator cannot be zero.

    $$x^2-12\times x+35\ne 0$$

26. A divisor fraction must also be nonzero, so its numerator cannot be zero.

    $$x^3-5\times x^2-14\times x\ne 0$$

27. Change division by this fraction into multiplication by its reciprocal.

    $$\frac{x^2-12\times x+35}{x^3-5\times x^2-14\times x}$$

28. Expand any powers or inner brackets before factoring.

    $$x^2-12\times x+35=x^2-12\times x+35$$

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

    $$(-5-7=-12,\left(-5\right)\times \left(-7\right)=35)$$

30. Rewrite the middle term as those two terms.

    $$x^2-5\times x-7\times x+35$$

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

    $$x\times \left(x-5\right)-7\times \left(x-5\right)$$

32. Factor out the common bracket.

    $$\left(x-7\right)\times \left(x-5\right)$$

33. The complete factored form is:

    $$\left(x-7\right)\times \left(x-5\right)$$

34. Expand any powers or inner brackets before factoring.

    $$x^3-5\times x^2-14\times x=x^3-5\times x^2-14\times x$$

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

    $$x^3-5\times x^2-14\times x=x\times \left(x^2-14-5\times x\right)$$

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

    $$(2-7=-5,2\times \left(-7\right)=-14)$$

37. Rewrite the middle term as those two terms.

    $$x^2+2\times x-7\times x-14$$

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

    $$x\times \left(x+2\right)-7\times \left(x+2\right)$$

39. Factor out the common bracket.

    $$\left(x-7\right)\times \left(x+2\right)$$

40. The complete factored form is:

    $$x\times \left(x-7\right)\times \left(x+2\right)$$

41. Multiply all numerator factors and all denominator factors.

    $$\frac{\left(x\times \left(x+4\right)\right)\times \left(\left(x-7\right)\times \left(x-5\right)\right)\times \left(\left(x-3\right)\times \left(x+2\right)\right)}{\left(x^2\times \left(x-3\right)\right)\times \left(\left(x-5\right)\times \left(x+4\right)\right)\times \left(x\times \left(x-7\right)\times \left(x+2\right)\right)}$$

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

    $$\left(x^2\times \left(x-3\right)\right)\times \left(\left(x-5\right)\times \left(x+4\right)\right)\times \left(x\times \left(x-7\right)\times \left(x+2\right)\right)\ne 0$$

43. Factor the numerator.

44. Expand any powers or inner brackets before factoring.

    $$\left(x\times \left(x+4\right)\right)\times \left(\left(x-7\right)\times \left(x-5\right)\right)\times \left(\left(x-3\right)\times \left(x+2\right)\right)=x^6-9\times x^5-11\times x^4+201\times x^3-62\times x^2-840\times x$$

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

    $$x^6-9\times x^5-11\times x^4+201\times x^3-62\times x^2-840\times x=x\times \left(x^5-840-62\times x-11\times x^3-9\times x^4+201\times x^2\right)$$

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

    $$\left(x\times \left(x+4\right)\right)\times \left(\left(x-7\right)\times \left(x-5\right)\right)\times \left(\left(x-3\right)\times \left(x+2\right)\right)=x\times \left(x-7\right)\times \left(x-5\right)\times \left(x-3\right)\times \left(x+2\right)\times \left(x+4\right)$$

47. The complete factored form is:

    $$x\times \left(x-7\right)\times \left(x-5\right)\times \left(x-3\right)\times \left(x+2\right)\times \left(x+4\right)$$

48. Factor the denominator.

49. Expand any powers or inner brackets before factoring.

    $$\left(x^2\times \left(x-3\right)\right)\times \left(\left(x-5\right)\times \left(x+4\right)\right)\times \left(x\times \left(x-7\right)\times \left(x+2\right)\right)=x^8-9\times x^7-11\times x^6+201\times x^5-62\times x^4-840\times x^3$$

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

    $$x^8-9\times x^7-11\times x^6+201\times x^5-62\times x^4-840\times x^3=x^3\times \left(x^5-840-62\times x-11\times x^3-9\times x^4+201\times x^2\right)$$

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

    $$\left(x^2\times \left(x-3\right)\right)\times \left(\left(x-5\right)\times \left(x+4\right)\right)\times \left(x\times \left(x-7\right)\times \left(x+2\right)\right)=x^3\times \left(x-7\right)\times \left(x-5\right)\times \left(x-3\right)\times \left(x+2\right)\times \left(x+4\right)$$

52. The complete factored form is:

    $$x^3\times \left(x-7\right)\times \left(x-5\right)\times \left(x-3\right)\times \left(x+2\right)\times \left(x+4\right)$$

53. Write the factored numerator over the factored denominator.

    $$\frac{x\times \left(x-7\right)\times \left(x-5\right)\times \left(x-3\right)\times \left(x+2\right)\times \left(x+4\right)}{x^3\times \left(x-7\right)\times \left(x-5\right)\times \left(x-3\right)\times \left(x+2\right)\times \left(x+4\right)}$$

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

    $$x^6-9\times x^5-11\times x^4+201\times x^3-62\times x^2-840\times x$$

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

    $$\frac{\frac{\left(x\times \left(x+4\right)\right)\times \left(\left(x-7\right)\times \left(x-5\right)\right)\times \left(\left(x-3\right)\times \left(x+2\right)\right)}{x^6-9\times x^5-11\times x^4+201\times x^3-62\times x^2-840\times x}}{\frac{\left(x^2\times \left(x-3\right)\right)\times \left(\left(x-5\right)\times \left(x+4\right)\right)\times \left(x\times \left(x-7\right)\times \left(x+2\right)\right)}{x^6-9\times x^5-11\times x^4+201\times x^3-62\times x^2-840\times x}}$$

56. Simplify the two divisions separately.

    $$\frac{1}{x^2}$$

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

    $$\frac{1}{x^2}$$

Answer

The final answer is shown below.