Mathematics 10 — Exercise Review · Q2 (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.

   $$x^2-6\times x+9\ne 0$$

2. Factor the numerator.

3. Expand any powers or inner brackets before factoring.

   $$x^2-7\times x+12=x^2-7\times x+12$$

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

   $$(-3-4=-7,\left(-3\right)\times \left(-4\right)=12)$$

5. Rewrite the middle term as those two terms.

   $$x^2-3\times x-4\times x+12$$

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

   $$x\times \left(x-3\right)-4\times \left(x-3\right)$$

7. Factor out the common bracket.

   $$\left(x-4\right)\times \left(x-3\right)$$

8. The complete factored form is:

   $$\left(x-4\right)\times \left(x-3\right)$$

9. Factor the denominator.

10. Expand any powers or inner brackets before factoring.

    $$x^2-6\times x+9=x^2-6\times x+9$$

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

    $$(-3-3=-6,\left(-3\right)\times \left(-3\right)=9)$$

12. Rewrite the middle term as those two terms.

    $$x^2-3\times x-3\times x+9$$

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

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

14. Factor out the common bracket.

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

15. The complete factored form is:

    $${\left(x-3\right)}^2$$

16. Write the factored numerator over the factored denominator.

    $$\frac{\left(x-4\right)\times \left(x-3\right)}{{\left(x-3\right)}^2}$$

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

    $$x-3$$

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

    $$\frac{\frac{x^2-7\times x+12}{x-3}}{\frac{x^2-6\times x+9}{x-3}}$$

19. Simplify the two divisions separately.

    $$\frac{x-4}{x-3}$$

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

    $$\frac{x-4}{x-3}$$

Answer

The final answer is shown below.