Mathematics 10 — Exercise 5.1 · Q1 (v)

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.

   $$4\times x^2+6\times x\times y\ne 0$$

2. Factor the numerator.

3. Expand any powers or inner brackets before factoring.

   $$4\times x^2-9\times y^2=4\times x^2-9\times y^2$$

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

   $$4\times x^2-9\times y^2=\left(2\times x-3\times y\right)\times \left(2\times x+3\times y\right)$$

5. The complete factored form is:

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

6. Factor the denominator.

7. Expand any powers or inner brackets before factoring.

   $$4\times x^2+6\times x\times y=4\times x^2+6\times x\times y$$

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

   $$4\times x^2+6\times x\times y=\left(2\times x\right)\times \left(2\times x+3\times y\right)$$

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

   $$4\times x^2+6\times x\times y=2\times x\times \left(2\times x+3\times y\right)$$

10. The complete factored form is:

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

11. Write the factored numerator over the factored denominator.

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

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

    $$2\times x+3\times y$$

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

    $$\frac{\frac{4\times x^2-9\times y^2}{2\times x+3\times y}}{\frac{4\times x^2+6\times x\times y}{2\times x+3\times y}}$$

14. Simplify the two divisions separately.

    $$\frac{2\times x-3\times y}{2\times x}$$

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

    $$\frac{2\times x-3\times y}{2\times x}$$

Answer

The final answer is shown below.