Mathematics 10 — Exercise 5.1 · Q1 (ix)

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.

   $$x^2+4\times x+4\ne 0$$

2. Factor the numerator.

3. Expand any powers or inner brackets before factoring.

   $$3\times x^2+6\times x=3\times x^2+6\times x$$

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

   $$3\times x^2+6\times x=\left(3\times x\right)\times \left(x+2\right)$$

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

   $$3\times x^2+6\times x=3\times x\times \left(x+2\right)$$

6. The complete factored form is:

   $$3\times x\times \left(x+2\right)$$

7. Factor the denominator.

8. Expand any powers or inner brackets before factoring.

   $$x^2+4\times x+4=x^2+4\times x+4$$

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

   $$(2+2=4,2\times 2=4)$$

10. Rewrite the middle term as those two terms.

    $$x^2+2\times x+2\times x+4$$

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

    $$x\times \left(x+2\right)+2\times \left(x+2\right)$$

12. Factor out the common bracket.

    $$\left(x+2\right)\left(x+2\right)$$

13. The complete factored form is:

    $${\left(x+2\right)}^2$$

14. Write the factored numerator over the factored denominator.

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

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

    $$x+2$$

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

    $$\frac{\frac{3\times x^2+6\times x}{x+2}}{\frac{x^2+4\times x+4}{x+2}}$$

17. Simplify the two divisions separately.

    $$\frac{3\times x}{x+2}$$

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

    $$\frac{3\times x}{x+2}$$

Answer

The final answer is shown below.