Mathematics 10 — Exercise 5.1 · Q1 (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.

   $$a\times c\times x+c\times x^2\ne 0$$

2. Factor the numerator.

3. Expand any powers or inner brackets before factoring.

   $$a\times b\times x+b\times x^2=a\times b\times x+b\times x^2$$

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

   $$a\times b\times x+b\times x^2=\left(b\times x\right)\times \left(x+a\right)$$

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

   $$a\times b\times x+b\times x^2=b\times x\times \left(a+x\right)$$

6. The complete factored form is:

   $$b\times x\times \left(a+x\right)$$

7. Factor the denominator.

8. Expand any powers or inner brackets before factoring.

   $$a\times c\times x+c\times x^2=a\times c\times x+c\times x^2$$

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

   $$a\times c\times x+c\times x^2=\left(c\times x\right)\times \left(x+a\right)$$

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

    $$a\times c\times x+c\times x^2=c\times x\times \left(a+x\right)$$

11. The complete factored form is:

    $$c\times x\times \left(a+x\right)$$

12. Write the factored numerator over the factored denominator.

    $$\frac{b\times x\times \left(a+x\right)}{c\times x\times \left(a+x\right)}$$

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

    $$a\times x+x^2$$

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

    $$\frac{\frac{a\times b\times x+b\times x^2}{a\times x+x^2}}{\frac{a\times c\times x+c\times x^2}{a\times x+x^2}}$$

15. Simplify the two divisions separately.

    $$\frac{b}{c}$$

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

    $$\frac{b}{c}$$

Answer

The final answer is shown below.