Mathematics 10 — Exercise Review · Q2 (ii)

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.

   $$7\times x-21\ne 0$$

2. Factor the numerator.

3. Expand any powers or inner brackets before factoring.

   $$x^2+3\times x-18=x^2+3\times x-18$$

4. Split the middle coefficient: these two numbers add to b and multiply to a·c.

   $$(6-3=3,6\times \left(-3\right)=-18)$$

5. Rewrite the middle term as those two terms.

   $$x^2+6\times x-3\times x-18$$

6. Group two terms at a time and take each common factor.

   $$x\times \left(x+6\right)-3\times \left(x+6\right)$$

7. Factor out the common bracket.

   $$\left(x-3\right)\times \left(x+6\right)$$

8. The complete factored form is:

   $$\left(x-3\right)\times \left(x+6\right)$$

9. Factor the denominator.

10. Expand any powers or inner brackets before factoring.

    $$7\times x-21=7\times x-21$$

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

    $$7\times x-21=7\times \left(x-3\right)$$

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

    $$7\times x-21=7\times \left(x-3\right)$$

13. The complete factored form is:

    $$7\times \left(x-3\right)$$

14. Write the factored numerator over the factored denominator.

    $$\frac{\left(x-3\right)\times \left(x+6\right)}{7\times \left(x-3\right)}$$

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

    $$x-3$$

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

    $$\frac{\frac{x^2+3\times x-18}{x-3}}{\frac{7\times x-21}{x-3}}$$

17. Simplify the two divisions separately.

    $$\frac{x+6}{7}$$

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

    $$\frac{x+6}{7}$$

Answer

The final answer is shown below.