Mathematics 10 — Exercise 5.1 · Q1 (iii)

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.

   $$a^2\times x^2-a\times x\ne 0$$

2. Factor the numerator.

3. Factor the denominator.

4. Expand any powers or inner brackets before factoring.

   $$a^2\times x^2-a\times x=a^2\times x^2-a\times x$$

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

   $$a^2\times x^2-a\times x=\left(a\times x\right)\times \left(a\times x-1\right)$$

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

   $$a^2\times x^2-a\times x=a\times x\times \left(a\times x-1\right)$$

7. The complete factored form is:

   $$a\times x\times \left(a\times x-1\right)$$

8. Write the factored numerator over the factored denominator.

   $$\frac{a\times c}{a\times x\times \left(a\times x-1\right)}$$

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

   $$a$$

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

    $$\frac{\frac{a\times c}{a}}{\frac{a^2\times x^2-a\times x}{a}}$$

11. Simplify the two divisions separately.

    $$\frac{c}{a\times x^2-x}$$

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

    $$\frac{c}{x\times \left(a\times x-1\right)}$$

Answer

The final answer is shown below.