Mathematics 10 — Exercise 5.3 · Q1 (viii)

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.

   $$10\times x^2+10\times x\times y-20\times y^2\ne 0$$

2. Expand any powers or inner brackets before factoring.

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

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

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

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

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

5. The complete factored form is:

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

6. Expand any powers or inner brackets before factoring.

   $$10\times x^2+10\times x\times y-20\times y^2=10\times x^2+10\times x\times y-20\times y^2$$

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

   $$10\times x^2+10\times x\times y-20\times y^2=10\times \left(x^2+x\times y-2\times y^2\right)$$

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

   $$10\times x^2+10\times x\times y-20\times y^2=10\times \left(x-y\right)\times \left(x+2\times y\right)$$

9. The complete factored form is:

   $$10\times \left(x-y\right)\times \left(x+2\times y\right)$$

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

    $$6\times x^2+6\times x\times y+6\times y^2\ne 0$$

11. Expand any powers or inner brackets before factoring.

    $$5\times x\times y+10\times y^2=5\times x\times y+10\times y^2$$

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

    $$5\times x\times y+10\times y^2=\left(5\times y\right)\times \left(2\times y+x\right)$$

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

    $$5\times x\times y+10\times y^2=5\times y\times \left(x+2\times y\right)$$

14. The complete factored form is:

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

15. Expand any powers or inner brackets before factoring.

    $$6\times x^2+6\times x\times y+6\times y^2=6\times x^2+6\times x\times y+6\times y^2$$

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

    $$6\times x^2+6\times x\times y+6\times y^2=6\times \left(x^2+x\times y+y^2\right)$$

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

    $$6\times x^2+6\times x\times y+6\times y^2=6\times \left(x^2+x\times y+y^2\right)$$

18. The complete factored form is:

    $$6\times \left(x^2+x\times y+y^2\right)$$

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

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

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

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

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

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

22. Expand any powers or inner brackets before factoring.

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

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

    $$4\times x^3-4\times y^3=4\times \left(x^3-y^3\right)$$

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

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

25. The complete factored form is:

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

26. Expand any powers or inner brackets before factoring.

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

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

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

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

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

29. The complete factored form is:

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

30. Multiply all numerator factors and all denominator factors.

    $$\frac{\left(3\times x\times \left(x-y\right)\right)\times \left(5\times y\times \left(x+2\times y\right)\right)\times \left(4\times \left(x-y\right)\times \left(x^2+x\times y+y^2\right)\right)}{\left(2\times \left(x-y\right)\times \left(x+y\right)\right)\times \left(10\times \left(x-y\right)\times \left(x+2\times y\right)\right)\times \left(6\times \left(x^2+x\times y+y^2\right)\right)}$$

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

    $$\left(2\times \left(x-y\right)\times \left(x+y\right)\right)\times \left(10\times \left(x-y\right)\times \left(x+2\times y\right)\right)\times \left(6\times \left(x^2+x\times y+y^2\right)\right)\ne 0$$

32. Factor the numerator.

33. Expand any powers or inner brackets before factoring.

    $$\left(3\times x\times \left(x-y\right)\right)\times \left(5\times y\times \left(x+2\times y\right)\right)\times \left(4\times \left(x-y\right)\times \left(x^2+x\times y+y^2\right)\right)=60\times x^6\times y+60\times x^5\times y^2-120\times x^4\times y^3-60\times x^3\times y^4-60\times x^2\times y^5+120\times x\times y^6$$

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

    $$60\times x^6\times y+60\times x^5\times y^2-120\times x^4\times y^3-60\times x^3\times y^4-60\times x^2\times y^5+120\times x\times y^6=\left(60\times x\times y\right)\times \left(x^5+x^4\times y-x\times y^4-x^2\times y^3-2\times x^3\times y^2+2\times y^5\right)$$

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

    $$\left(3\times x\times \left(x-y\right)\right)\times \left(5\times y\times \left(x+2\times y\right)\right)\times \left(4\times \left(x-y\right)\times \left(x^2+x\times y+y^2\right)\right)=60\times x\times y\times {\left(x-y\right)}^2\times \left(x+2\times y\right)\times \left(x^2+x\times y+y^2\right)$$

36. The complete factored form is:

    $$60\times x\times y\times {\left(x-y\right)}^2\times \left(x+2\times y\right)\times \left(x^2+x\times y+y^2\right)$$

37. Factor the denominator.

38. Expand any powers or inner brackets before factoring.

    $$\left(2\times \left(x-y\right)\times \left(x+y\right)\right)\times \left(10\times \left(x-y\right)\times \left(x+2\times y\right)\right)\times \left(6\times \left(x^2+x\times y+y^2\right)\right)=120\times x^6+240\times x^5\times y-120\times x^4\times y^2-360\times x^3\times y^3-240\times x^2\times y^4+120\times x\times y^5+240\times y^6$$

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

    $$120\times x^6+240\times x^5\times y-120\times x^4\times y^2-360\times x^3\times y^3-240\times x^2\times y^4+120\times x\times y^5+240\times y^6=120\times \left(x^6+2\times x^5\times y-x^4\times y^2-3\times x^3\times y^3-2\times x^2\times y^4+x\times y^5+2\times y^6\right)$$

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

    $$\left(2\times \left(x-y\right)\times \left(x+y\right)\right)\times \left(10\times \left(x-y\right)\times \left(x+2\times y\right)\right)\times \left(6\times \left(x^2+x\times y+y^2\right)\right)=120\times {\left(x-y\right)}^2\times \left(x+y\right)\times \left(x+2\times y\right)\times \left(x^2+x\times y+y^2\right)$$

41. The complete factored form is:

    $$120\times {\left(x-y\right)}^2\times \left(x+y\right)\times \left(x+2\times y\right)\times \left(x^2+x\times y+y^2\right)$$

42. Write the factored numerator over the factored denominator.

    $$\frac{60\times x\times y\times {\left(x-y\right)}^2\times \left(x+2\times y\right)\times \left(x^2+x\times y+y^2\right)}{120\times {\left(x-y\right)}^2\times \left(x+y\right)\times \left(x+2\times y\right)\times \left(x^2+x\times y+y^2\right)}$$

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

    $$60\times x^5+60\times x^4\times y-120\times x^3\times y^2-60\times x^2\times y^3-60\times x\times y^4+120\times y^5$$

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

    $$\frac{\frac{\left(3\times x\times \left(x-y\right)\right)\times \left(5\times y\times \left(x+2\times y\right)\right)\times \left(4\times \left(x-y\right)\times \left(x^2+x\times y+y^2\right)\right)}{60\times x^5+60\times x^4\times y-120\times x^3\times y^2-60\times x^2\times y^3-60\times x\times y^4+120\times y^5}}{\frac{\left(2\times \left(x-y\right)\times \left(x+y\right)\right)\times \left(10\times \left(x-y\right)\times \left(x+2\times y\right)\right)\times \left(6\times \left(x^2+x\times y+y^2\right)\right)}{60\times x^5+60\times x^4\times y-120\times x^3\times y^2-60\times x^2\times y^3-60\times x\times y^4+120\times y^5}}$$

45. Simplify the two divisions separately.

    $$\frac{x\times y}{2\times x+2\times y}$$

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

    $$\frac{x\times y}{2\times x+2\times y}$$

Answer

The final answer is shown below.