Mathematics 10 — Exercise 5.1 · Q1 (viii)

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.

   $$x^2-4\times x-5\ne 0$$

2. Factor the numerator.

3. Expand any powers or inner brackets before factoring.

   $$x^2-5\times x=x^2-5\times x$$

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

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

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

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

6. The complete factored form is:

   $$x\times \left(x-5\right)$$

7. Factor the denominator.

8. Expand any powers or inner brackets before factoring.

   $$x^2-4\times x-5=x^2-4\times x-5$$

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

   $$(1-5=-4,1\times \left(-5\right)=-5)$$

10. Rewrite the middle term as those two terms.

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

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

    $$x\times \left(x+1\right)-5\times \left(x+1\right)$$

12. Factor out the common bracket.

    $$\left(x-5\right)\times \left(x+1\right)$$

13. The complete factored form is:

    $$\left(x-5\right)\times \left(x+1\right)$$

14. Write the factored numerator over the factored denominator.

    $$\frac{x\times \left(x-5\right)}{\left(x-5\right)\times \left(x+1\right)}$$

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

    $$x-5$$

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

    $$\frac{\frac{x^2-5\times x}{x-5}}{\frac{x^2-4\times x-5}{x-5}}$$

17. Simplify the two divisions separately.

    $$\frac{x}{x+1}$$

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

    $$\frac{x}{x+1}$$

Answer

The final answer is shown below.