Mathematics 10 — Exercise 5.1 · Q1 (i)

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.

   $$2\times a^2\times b-4\times a\times b^2\ne 0$$

2. Factor the numerator.

3. Expand any powers or inner brackets before factoring.

   $$3\times a^2-6\times a\times b=3\times a^2-6\times a\times b$$

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

   $$3\times a^2-6\times a\times b=\left(3\times a\right)\times \left(a-2\times b\right)$$

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

   $$3\times a^2-6\times a\times b=3\times a\times \left(a-2\times b\right)$$

6. The complete factored form is:

   $$3\times a\times \left(a-2\times b\right)$$

7. Factor the denominator.

8. Expand any powers or inner brackets before factoring.

   $$2\times a^2\times b-4\times a\times b^2=2\times a^2\times b-4\times a\times b^2$$

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

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

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

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

11. The complete factored form is:

    $$2\times a\times b\times \left(a-2\times b\right)$$

12. Write the factored numerator over the factored denominator.

    $$\frac{3\times a\times \left(a-2\times b\right)}{2\times a\times b\times \left(a-2\times b\right)}$$

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

    $$a^2-2\times a\times b$$

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

    $$\frac{\frac{3\times a^2-6\times a\times b}{a^2-2\times a\times b}}{\frac{2\times a^2\times b-4\times a\times b^2}{a^2-2\times a\times b}}$$

15. Simplify the two divisions separately.

    $$\frac{3}{2\times b}$$

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

    $$\frac{3}{2\times b}$$

Answer

The final answer is shown below.