Mathematics 10 — Exercise 5.1 · Q1 (vi)

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.

   $$5\times x^2+5\times x\times y+5\times y^2\ne 0$$

2. Factor the numerator.

3. Expand any powers or inner brackets before factoring.

   $$20\times x^3-20\times y^3=20\times x^3-20\times y^3$$

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

   $$20\times x^3-20\times y^3=20\times \left(x^3-y^3\right)$$

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

   $$20\times x^3-20\times y^3=20\times \left(x-y\right)\times \left(x^2+x\times y+y^2\right)$$

6. The complete factored form is:

   $$20\times \left(x-y\right)\times \left(x^2+x\times y+y^2\right)$$

7. Factor the denominator.

8. Expand any powers or inner brackets before factoring.

   $$5\times x^2+5\times x\times y+5\times y^2=5\times x^2+5\times x\times y+5\times y^2$$

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

   $$5\times x^2+5\times x\times y+5\times y^2=5\times \left(x^2+x\times y+y^2\right)$$

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

    $$5\times x^2+5\times x\times y+5\times y^2=5\times \left(x^2+x\times y+y^2\right)$$

11. The complete factored form is:

    $$5\times \left(x^2+x\times y+y^2\right)$$

12. Write the factored numerator over the factored denominator.

    $$\frac{20\times \left(x-y\right)\times \left(x^2+x\times y+y^2\right)}{5\times \left(x^2+x\times y+y^2\right)}$$

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

    $$5\times x^2+5\times x\times y+5\times y^2$$

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

    $$\frac{\frac{20\times x^3-20\times y^3}{5\times x^2+5\times x\times y+5\times y^2}}{\frac{5\times x^2+5\times x\times y+5\times y^2}{5\times x^2+5\times x\times y+5\times y^2}}$$

15. Simplify the two divisions separately.

    $$\frac{1}{1}\times \left(4\times x-4\times y\right)$$

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

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

Answer

The final answer is shown below.