Mathematics 10 — Exercise 5.3 · Q1 (ii)

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.

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

2. Expand any powers or inner brackets before factoring.

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

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

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

4. Rewrite the middle term as those two terms.

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

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

   $$x\times \left(x-1\right)-2\times \left(x-1\right)$$

6. Factor out the common bracket.

   $$\left(x-2\right)\times \left(x-1\right)$$

7. The complete factored form is:

   $$\left(x-2\right)\times \left(x-1\right)$$

8. Expand any powers or inner brackets before factoring.

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

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

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

10. Rewrite the middle term as those two terms.

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

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

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

12. Factor out the common bracket.

    $$\left(x-1\right)\times \left(x+4\right)$$

13. The complete factored form is:

    $$\left(x-1\right)\times \left(x+4\right)$$

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

    $$3\times x+6\ne 0$$

15. Expand any powers or inner brackets before factoring.

    $$2\times x^2+8\times x=2\times x^2+8\times x$$

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

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

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

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

18. The complete factored form is:

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

19. Expand any powers or inner brackets before factoring.

    $$3\times x+6=3\times x+6$$

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

    $$3\times x+6=3\times \left(x+2\right)$$

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

    $$3\times x+6=3\times \left(x+2\right)$$

22. The complete factored form is:

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

23. Multiply all numerator factors and all denominator factors.

    $$\frac{\left(2\times x\times \left(x+4\right)\right)\times \left(\left(x-2\right)\times \left(x-1\right)\right)}{\left(\left(x-1\right)\times \left(x+4\right)\right)\times \left(3\times \left(x+2\right)\right)}$$

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

    $$\left(\left(x-1\right)\times \left(x+4\right)\right)\times \left(3\times \left(x+2\right)\right)\ne 0$$

25. Factor the numerator.

26. Expand any powers or inner brackets before factoring.

    $$\left(2\times x\times \left(x+4\right)\right)\times \left(\left(x-2\right)\times \left(x-1\right)\right)=2\times x^4+2\times x^3-20\times x^2+16\times x$$

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

    $$2\times x^4+2\times x^3-20\times x^2+16\times x=\left(2\times x\right)\times \left(x^2+x^3-10\times x+8\right)$$

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

    $$\left(2\times x\times \left(x+4\right)\right)\times \left(\left(x-2\right)\times \left(x-1\right)\right)=2\times x\times \left(x-2\right)\times \left(x-1\right)\times \left(x+4\right)$$

29. The complete factored form is:

    $$2\times x\times \left(x-2\right)\times \left(x-1\right)\times \left(x+4\right)$$

30. Factor the denominator.

31. Expand any powers or inner brackets before factoring.

    $$\left(\left(x-1\right)\times \left(x+4\right)\right)\times \left(3\times \left(x+2\right)\right)=3\times x^3+15\times x^2+6\times x-24$$

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

    $$3\times x^3+15\times x^2+6\times x-24=3\times \left(x^3+5\times x^2+2\times x-8\right)$$

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

    $$\left(\left(x-1\right)\times \left(x+4\right)\right)\times \left(3\times \left(x+2\right)\right)=3\times \left(x-1\right)\times \left(x+2\right)\times \left(x+4\right)$$

34. The complete factored form is:

    $$3\times \left(x-1\right)\times \left(x+2\right)\times \left(x+4\right)$$

35. Write the factored numerator over the factored denominator.

    $$\frac{2\times x\times \left(x-2\right)\times \left(x-1\right)\times \left(x+4\right)}{3\times \left(x-1\right)\times \left(x+2\right)\times \left(x+4\right)}$$

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

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

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

    $$\frac{\frac{\left(2\times x\times \left(x+4\right)\right)\times \left(\left(x-2\right)\times \left(x-1\right)\right)}{x^2+3\times x-4}}{\frac{\left(\left(x-1\right)\times \left(x+4\right)\right)\times \left(3\times \left(x+2\right)\right)}{x^2+3\times x-4}}$$

38. Simplify the two divisions separately.

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

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

    $$\frac{2\times x\times \left(x-2\right)}{3\times x+6}$$

Answer

The final answer is shown below.