Mathematics 10 — Exercise 5.2 · Q1 (xi)

Full question

Simplify the following sum or difference of algebraic fractions.

Step-by-step solution

1. Fraction 1: keep its denominator nonzero and factor that denominator.

   $$x-y\ne 0$$

2. Fraction 2: keep its denominator nonzero and factor that denominator.

   $$-x+y\ne 0$$

3. Build the least common denominator from every distinct denominator factor at its largest required power.

   $$-x+y$$

4. For fraction 1, divide the common denominator by its own denominator.

   $$\frac{-x+y}{x-y}=-1$$

5. Multiply this numerator by the same factor to keep the fraction unchanged.

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

6. For fraction 2, divide the common denominator by its own denominator.

   $$\frac{-x+y}{-x+y}=1$$

7. Multiply this numerator by the same factor to keep the fraction unchanged.

   $$\frac{1\times y^3}{-x+y}$$

8. The denominators now match. Put all signed numerators over one common denominator.

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

9. Expand numerator product 1, keeping minus signs on the whole bracket.

   $$\left(-1\right)\times x^3=-x^3$$

10. Expand numerator product 2, keeping minus signs on the whole bracket.

    $$1\times y^3=y^3$$

11. Collect like powers and constants in the combined numerator.

    $$-x^3+y^3=-x^3+y^3$$

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

    $$-x+y\ne 0$$

13. Factor the numerator.

14. Expand any powers or inner brackets before factoring.

    $$-x^3+y^3=-x^3+y^3$$

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

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

16. The complete factored form is:

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

17. Factor the denominator.

18. Write the factored numerator over the factored denominator.

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

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

    $$x-y$$

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

    $$\frac{\frac{-x^3+y^3}{x-y}}{\frac{-x+y}{x-y}}$$

21. Simplify the two divisions separately.

    $$\frac{-x^2-x\times y-y^2}{-1}$$

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

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

Answer

The final answer is shown below.