Mathematics 10 — Exercise 5.2 · Q2

Full question

Subtract 1/(x²+2) FROM (2x³+x²+3)/(x²+2)².

Step-by-step solution

1. Fraction 1: keep its denominator nonzero and factor that denominator.

   $${\left(x^2+2\right)}^2\ne 0$$

2. Fraction 2: keep its denominator nonzero and factor that denominator.

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

3. Build the least common denominator from every distinct denominator factor at its largest required power.

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

4. For fraction 1, divide the common denominator by its own denominator.

   $$\frac{x^4+4\times x^2+4}{{\left(x^2+2\right)}^2}=1$$

5. Multiply this numerator by the same factor to keep the fraction unchanged.

   $$\frac{1\times \left(2\times x^3+x^2+3\right)}{{\left(x^2+2\right)}^2}$$

6. For fraction 2, divide the common denominator by its own denominator.

   $$\frac{x^4+4\times x^2+4}{x^2+2}=x^2+2$$

7. Multiply this numerator by the same factor to keep the fraction unchanged.

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

8. The denominators now match. Put all signed numerators over one common denominator.

   $$\frac{1\times \left(2\times x^3+x^2+3\right)-(x^2+2)}{{\left(x^2+2\right)}^2}$$

9. Expand numerator product 1, keeping minus signs on the whole bracket.

   $$1\times \left(2\times x^3+x^2+3\right)=2\times x^3+x^2+3$$

10. Expand numerator product 2, keeping minus signs on the whole bracket.

    $$-(x^2+2)=-x^2-2$$

11. Collect like powers and constants in the combined numerator.

    $$2\times x^3+x^2+3-(x^2+2)=2\times x^3+1$$

12. The original denominator must be nonzero. Keep this restriction even if a factor is later cancelled.

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

13. Factor the numerator.

14. Factor the denominator.

15. Expand any powers or inner brackets before factoring.

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

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

    $$x^4+4\times x^2+4={\left(x^2+2\right)}^2$$

17. The complete factored form is:

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

18. Write the factored numerator over the factored denominator.

    $$\frac{2\times x^3+1}{{\left(x^2+2\right)}^2}$$

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

    $$\frac{2\times x^3+1}{{\left(x^2+2\right)}^2}$$

Answer

The final answer is shown below.