Mathematics 10 — Exercise 5.3 · Q1 (v)

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.

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

2. Expand any powers or inner brackets before factoring.

   $$x^3-8=x^3-8$$

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

   $$x^3-8=\left(x-2\right)\times \left(x^2+2\times x+4\right)$$

4. The complete factored form is:

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

5. Expand any powers or inner brackets before factoring.

   $$x^2-4=x^2-4$$

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

   $$(2-2=0,2\times \left(-2\right)=-4)$$

7. Rewrite the middle term as those two terms.

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

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

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

9. Factor out the common bracket.

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

10. The complete factored form is:

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

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

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

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

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

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

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

14. Expand any powers or inner brackets before factoring.

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

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

    $$(2+2=4,2\times 2=4)$$

16. Rewrite the middle term as those two terms.

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

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

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

18. Factor out the common bracket.

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

19. The complete factored form is:

    $${\left(x+2\right)}^2$$

20. Multiply all numerator factors and all denominator factors.

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

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

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

22. Factor the numerator.

23. Expand any powers or inner brackets before factoring.

    $$\left(\left(x-2\right)\times \left(x^2+2\times x+4\right)\right)\times {\left(x+2\right)}^2=x^5+4\times x^4+4\times x^3-8\times x^2-32\times x-32$$

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

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

25. The complete factored form is:

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

26. Factor the denominator.

27. Expand any powers or inner brackets before factoring.

    $$\left(\left(x-2\right)\times \left(x+2\right)\right)\times \left(x^2+2\times x+4\right)=x^4+2\times x^3-8\times x-16$$

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

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

29. The complete factored form is:

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

30. Write the factored numerator over the factored denominator.

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

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

    $$x^4+2\times x^3-8\times x-16$$

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

    $$\frac{\frac{\left(\left(x-2\right)\times \left(x^2+2\times x+4\right)\right)\times {\left(x+2\right)}^2}{x^4+2\times x^3-8\times x-16}}{\frac{\left(\left(x-2\right)\times \left(x+2\right)\right)\times \left(x^2+2\times x+4\right)}{x^4+2\times x^3-8\times x-16}}$$

33. Simplify the two divisions separately.

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

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

    $$x+2$$

Answer

The final answer is shown below.