Mathematics 10 — Exercise 5.3 · Q1 (ix)

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.

   $$x^3-125\ne 0$$

2. Expand any powers or inner brackets before factoring.

   $$2\times x^2-98=2\times x^2-98$$

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

   $$2\times x^2-98=2\times \left(x^2-49\right)$$

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

   $$(7-7=0,7\times \left(-7\right)=-49)$$

5. Rewrite the middle term as those two terms.

   $$x^2+7\times x-7\times x-49$$

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

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

7. Factor out the common bracket.

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

8. The complete factored form is:

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

9. Expand any powers or inner brackets before factoring.

   $$x^3-125=x^3-125$$

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

    $$x^3-125=\left(x-5\right)\times \left(x^2+5\times x+25\right)$$

11. The complete factored form is:

    $$\left(x-5\right)\times \left(x^2+5\times x+25\right)$$

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

    $$3\times x-21\ne 0$$

13. Expand any powers or inner brackets before factoring.

    $$x^2-3\times x-10=x^2-3\times x-10$$

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

    $$(2-5=-3,2\times \left(-5\right)=-10)$$

15. Rewrite the middle term as those two terms.

    $$x^2+2\times x-5\times x-10$$

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

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

17. Factor out the common bracket.

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

18. The complete factored form is:

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

19. Expand any powers or inner brackets before factoring.

    $$3\times x-21=3\times x-21$$

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

    $$3\times x-21=3\times \left(x-7\right)$$

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

    $$3\times x-21=3\times \left(x-7\right)$$

22. The complete factored form is:

    $$3\times \left(x-7\right)$$

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

    $$x^2+5\times x+25\ne 0$$

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

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

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

    $$\frac{x^2+5\times x+25}{x^2+5\times x-14}$$

26. Expand any powers or inner brackets before factoring.

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

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

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

28. Rewrite the middle term as those two terms.

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

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

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

30. Factor out the common bracket.

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

31. The complete factored form is:

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

32. Multiply all numerator factors and all denominator factors.

    $$\frac{\left(2\times \left(x-7\right)\times \left(x+7\right)\right)\times \left(\left(x-5\right)\times \left(x+2\right)\right)\times \left(x^2+5\times x+25\right)}{\left(\left(x-5\right)\times \left(x^2+5\times x+25\right)\right)\times \left(\left(x-2\right)\times \left(x+7\right)\right)\times \left(3\times \left(x-7\right)\right)}$$

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

    $$\left(\left(x-5\right)\times \left(x^2+5\times x+25\right)\right)\times \left(\left(x-2\right)\times \left(x+7\right)\right)\times \left(3\times \left(x-7\right)\right)\ne 0$$

34. Factor the numerator.

35. Expand any powers or inner brackets before factoring.

    $$\left(2\times \left(x-7\right)\times \left(x+7\right)\right)\times \left(\left(x-5\right)\times \left(x+2\right)\right)\times \left(x^2+5\times x+25\right)=2\times x^6+4\times x^5-98\times x^4-446\times x^3-500\times x^2+12250\times x+24500$$

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

    $$2\times x^6+4\times x^5-98\times x^4-446\times x^3-500\times x^2+12250\times x+24500=2\times \left(x^6+2\times x^5-49\times x^4-223\times x^3-250\times x^2+6125\times x+12250\right)$$

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

    $$\left(2\times \left(x-7\right)\times \left(x+7\right)\right)\times \left(\left(x-5\right)\times \left(x+2\right)\right)\times \left(x^2+5\times x+25\right)=2\times \left(x-7\right)\times \left(x-5\right)\times \left(x+2\right)\times \left(x+7\right)\times \left(x^2+5\times x+25\right)$$

38. The complete factored form is:

    $$2\times \left(x-7\right)\times \left(x-5\right)\times \left(x+2\right)\times \left(x+7\right)\times \left(x^2+5\times x+25\right)$$

39. Factor the denominator.

40. Expand any powers or inner brackets before factoring.

    $$\left(\left(x-5\right)\times \left(x^2+5\times x+25\right)\right)\times \left(\left(x-2\right)\times \left(x+7\right)\right)\times \left(3\times \left(x-7\right)\right)=3\times x^6-6\times x^5-147\times x^4-81\times x^3+750\times x^2+18375\times x-36750$$

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

    $$3\times x^6-6\times x^5-147\times x^4-81\times x^3+750\times x^2+18375\times x-36750=3\times \left(x^6-2\times x^5-49\times x^4-27\times x^3+250\times x^2+6125\times x-12250\right)$$

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

    $$\left(\left(x-5\right)\times \left(x^2+5\times x+25\right)\right)\times \left(\left(x-2\right)\times \left(x+7\right)\right)\times \left(3\times \left(x-7\right)\right)=3\times \left(x-7\right)\times \left(x-5\right)\times \left(x-2\right)\times \left(x+7\right)\times \left(x^2+5\times x+25\right)$$

43. The complete factored form is:

    $$3\times \left(x-7\right)\times \left(x-5\right)\times \left(x-2\right)\times \left(x+7\right)\times \left(x^2+5\times x+25\right)$$

44. Write the factored numerator over the factored denominator.

    $$\frac{2\times \left(x-7\right)\times \left(x-5\right)\times \left(x+2\right)\times \left(x+7\right)\times \left(x^2+5\times x+25\right)}{3\times \left(x-7\right)\times \left(x-5\right)\times \left(x-2\right)\times \left(x+7\right)\times \left(x^2+5\times x+25\right)}$$

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

    $$x^5-49\times x^3-125\times x^2+6125$$

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

    $$\frac{\frac{\left(2\times \left(x-7\right)\times \left(x+7\right)\right)\times \left(\left(x-5\right)\times \left(x+2\right)\right)\times \left(x^2+5\times x+25\right)}{x^5-49\times x^3-125\times x^2+6125}}{\frac{\left(\left(x-5\right)\times \left(x^2+5\times x+25\right)\right)\times \left(\left(x-2\right)\times \left(x+7\right)\right)\times \left(3\times \left(x-7\right)\right)}{x^5-49\times x^3-125\times x^2+6125}}$$

47. Simplify the two divisions separately.

    $$\frac{2\times x+4}{3\times x-6}$$

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

    $$\frac{2\times x+4}{3\times x-6}$$

Answer

The final answer is shown below.