Mathematics 10 — Exercise 5.1 · Q1 (x)

Full question

Reduce the following algebraic fraction to its lowest terms. State the values of the variables for which the original denominator is zero and exclude them.

Step-by-step solution

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

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

2. Factor the numerator.

3. Expand any powers or inner brackets before factoring.

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

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

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

5. The complete factored form is:

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

6. Factor the denominator.

7. Expand any powers or inner brackets before factoring.

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

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

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

9. The complete factored form is:

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

10. Write the factored numerator over the factored denominator.

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

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

    $$x-y$$

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

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

13. Simplify the two divisions separately.

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

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

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

Answer

The final answer is shown below.