Mathematics 10 — Exercise Review · Q3 (iii)

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.

   $$-c^2+{\left(a+b\right)}^2\ne 0$$

2. Expand any powers or inner brackets before factoring.

   $$a^2-{\left(b-c\right)}^2=a^2-b^2+2\times b\times c-c^2$$

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

   $$a^2-{\left(b-c\right)}^2=\left(a-b+c\right)\times \left(a+b-c\right)$$

4. The complete factored form is:

   $$\left(a-b+c\right)\times \left(a+b-c\right)$$

5. Expand any powers or inner brackets before factoring.

   $$-c^2+{\left(a+b\right)}^2=a^2+2\times a\times b+b^2-c^2$$

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

   $$-c^2+{\left(a+b\right)}^2=\left(a+b-c\right)\times \left(a+b+c\right)$$

7. The complete factored form is:

   $$\left(a+b-c\right)\times \left(a+b+c\right)$$

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

   $$-c^2+{\left(a-b\right)}^2\ne 0$$

9. Expand any powers or inner brackets before factoring.

   $$a^2-{\left(b+c\right)}^2=a^2-b^2-2\times b\times c-c^2$$

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

    $$a^2-{\left(b+c\right)}^2=\left(a-b-c\right)\times \left(a+b+c\right)$$

11. The complete factored form is:

    $$\left(a-b-c\right)\times \left(a+b+c\right)$$

12. Expand any powers or inner brackets before factoring.

    $$-c^2+{\left(a-b\right)}^2=a^2-2\times a\times b+b^2-c^2$$

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

    $$-c^2+{\left(a-b\right)}^2=\left(a-b-c\right)\times \left(a-b+c\right)$$

14. The complete factored form is:

    $$\left(a-b-c\right)\times \left(a-b+c\right)$$

15. Multiply all numerator factors and all denominator factors.

    $$\frac{\left(\left(a-b-c\right)\times \left(a+b+c\right)\right)\times \left(\left(a-b+c\right)\times \left(a+b-c\right)\right)}{\left(\left(a-b-c\right)\times \left(a-b+c\right)\right)\times \left(\left(a+b-c\right)\times \left(a+b+c\right)\right)}$$

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

    $$\left(\left(a-b-c\right)\times \left(a-b+c\right)\right)\times \left(\left(a+b-c\right)\times \left(a+b+c\right)\right)\ne 0$$

17. Factor the numerator.

18. Expand any powers or inner brackets before factoring.

    $$\left(\left(a-b-c\right)\times \left(a+b+c\right)\right)\times \left(\left(a-b+c\right)\times \left(a+b-c\right)\right)=a^4-2\times a^2\times b^2-2\times a^2\times c^2+b^4-2\times b^2\times c^2+c^4$$

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

    $$\left(\left(a-b-c\right)\times \left(a+b+c\right)\right)\times \left(\left(a-b+c\right)\times \left(a+b-c\right)\right)=\left(a-b-c\right)\times \left(a-b+c\right)\times \left(a+b-c\right)\times \left(a+b+c\right)$$

20. The complete factored form is:

    $$\left(a-b-c\right)\times \left(a-b+c\right)\times \left(a+b-c\right)\times \left(a+b+c\right)$$

21. Factor the denominator.

22. Expand any powers or inner brackets before factoring.

    $$\left(\left(a-b-c\right)\times \left(a-b+c\right)\right)\times \left(\left(a+b-c\right)\times \left(a+b+c\right)\right)=a^4-2\times a^2\times b^2-2\times a^2\times c^2+b^4-2\times b^2\times c^2+c^4$$

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

    $$\left(\left(a-b-c\right)\times \left(a-b+c\right)\right)\times \left(\left(a+b-c\right)\times \left(a+b+c\right)\right)=\left(a-b-c\right)\times \left(a-b+c\right)\times \left(a+b-c\right)\times \left(a+b+c\right)$$

24. The complete factored form is:

    $$\left(a-b-c\right)\times \left(a-b+c\right)\times \left(a+b-c\right)\times \left(a+b+c\right)$$

25. Write the factored numerator over the factored denominator.

    $$\frac{\left(a-b-c\right)\times \left(a-b+c\right)\times \left(a+b-c\right)\times \left(a+b+c\right)}{\left(a-b-c\right)\times \left(a-b+c\right)\times \left(a+b-c\right)\times \left(a+b+c\right)}$$

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

    $$a^4-2\times a^2\times b^2-2\times a^2\times c^2+b^4-2\times b^2\times c^2+c^4$$

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

    $$\frac{\frac{\left(\left(a-b-c\right)\times \left(a+b+c\right)\right)\times \left(\left(a-b+c\right)\times \left(a+b-c\right)\right)}{a^4-2\times a^2\times b^2-2\times a^2\times c^2+b^4-2\times b^2\times c^2+c^4}}{\frac{\left(\left(a-b-c\right)\times \left(a-b+c\right)\right)\times \left(\left(a+b-c\right)\times \left(a+b+c\right)\right)}{a^4-2\times a^2\times b^2-2\times a^2\times c^2+b^4-2\times b^2\times c^2+c^4}}$$

28. Simplify the two divisions separately.

    $$\frac{1}{1}$$

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

    $$1$$

Answer

The final answer is shown below.