Rational Expressions (AQA A-Level Mathematics): Revision Notes

Example: $\dfrac{x^2 + 5}{x - 8}$ is a rational function.

Example 1:

1. Factorize the polynomials:

1. Cancel common factors:

Result:

Example 2:

2. Find a common denominator:

2. Combine into a single fraction:

3. Simplify the expression:

Example 3: Simplify fully

3. Factorize both numerator and denominator:

4. Cancel common factors:

Final result:

Express $\dfrac{9x^2 + 20x - 10}{(x+2)(3x-1)}$ in partial fractions. Notice that the denominator has the same order as the numerator. This means the division algorithm will give rise to a constant term.

Do division by grid method:

5. Divide $9x^2$ by $3x^2$ to get $3$ (Quotient):

6. Subtract to find the remainder:

Key Point: If the numerator is top-heavy, i.e., has a degree $\geq$ the denominator, a division must be performed before partial fractions can be found.

Concentrating on just the partial fractions and ignoring the quotient:

Multiply both sides by $(x+2)(3x-1)$ to get:

• Let $x = -2$:

• Let $x = \frac{1}{3}$:

GCSE Recap:

Write as a single fraction: $\dfrac{3}{x+2} + \dfrac{4}{x-3}$

Need to have the same denominator:

New Material:

Write $\dfrac{7x-1}{(x+2)(x-3)}$ as the sum of separate fractions.

7. Write an identity with the LHS as our combined fraction and the RHS as our target form:

8. Multiply both sides by the denominator of the LHS:

9. Substitute in 'tactical' values of $x$ to help find the values of $A$ and $B$:

• Let $x = 3$:

• Let $x = -2$:

Thus:

Example 1:

Write $\dfrac{2}{(x+1)(x+3)}$ as partial fractions.

10. Start by expressing $\dfrac{2}{(x+1)(x+3)}$ as the sum of two fractions:

11. Multiply through by the common denominator $(x+1)(x+3)$ to clear the fractions:

12. Substitute values of $x$ to solve for $A$ and $B$:

• Let $x = -3$:

• Let $x = -1$:

13. Therefore, the partial fraction decomposition is:

Example 2:

Express $\dfrac{1}{x^2 - 1}$ as partial fractions.

14. Start by expressing $\dfrac{1}{x^2 - 1}$ using the difference of squares:

15. Write the partial fractions decomposition:

16. Multiply both sides by the common denominator $(x+1)(x-1)$:

17. Substitute values of $x$ to solve for $A$ and $B$:

• Let $x = 1$:

• Let $x = -1$:

18. Therefore, the partial fraction decomposition is:

Example 3:

Express $\dfrac{9x-14}{(x+4)(x-1)^2}$ as partial fractions.

19. Setup Partial Fractions:

• Write the fraction as a sum of simpler fractions:

• Here, $(x-1)^2$ in the denominator leads to factors of $(x-1)$ and $(x-1)^2$.

20. Combine into a Single Expression:

21. Find Coefficients $A$, $B$, and $C$ by Substitution:

• Let $x = 1$:

• Let $x = -4$:

• For another value, let $x = 0$ (a non-clever substitution):

Substitute $A = -2$ and $C = -1$ into the equation:

22. Final Partial Fraction Decomposition:

Q5 (Jun 2010, Q3)

Express $\dfrac{x^2}{(x-1)^2(x-2)}$ in partial fractions.

23. Setup Partial Fractions:

• Write the fraction as a sum of simpler fractions:

24. Combine into a Single Expression:

25. Find Coefficients A, B, and C by Substitution:

• Let $x = 2$:

• Let $x = 1$:

• For another value, let $x = 0$ (to find $B$):

26. Final Partial Fraction Decomposition:

Q9 (Jun 2015, Q1) (i) Express $\dfrac{2}{3-x} + \frac{3}{1+x}$ as a single fraction in its simplest form.

Solution:

(ii) Hence express $\left(\dfrac{2}{3-x} + \dfrac{3}{1+x}\right) \times \dfrac{x^2 + 8x - 33}{121 - x^2}$ as a single fraction in its lowest terms.

Solution: