1. Module :MA0001NP Foundation Mathematics Lecture Week 6

2. Rational Expressions P, Q, R, and S are polynomials Addition Operation Multiplication Subtraction Division Notice the common denominator Reciprocal and Multiply

3. Addition Simplify

4. Addition Simplify

5. Subtraction Simplify

6. Subtraction Simplify 4

7. Multiplication Simplify

8. Multiplication Simplify Factorising (2x-6) 3

9. Division Simplify 2 2

10. Solve

12. Partial fractions Programme F7: Partial fractions Consider the following combination of algebraic fractions: The fractions on the left are called the partial fractions of the fraction on the right.

13. Partial fractions Find the partial fractions of the following Programme F7: Partial fractions

14. Partial fractions It is assumed that a partial fraction break down is possible in the form: The assumption is validated by finding the values of A and B. Partial fractions

15. To find the values of A and B the two partial fractions are added to give:

16. Partial fractions Programme F7: Partial fractions Since: And since the denominators are identical the numerators must be identical as well. That is:

17. Partial fractions Programme F7: Partial fractions Consider the identity: Therefore: Therefore Therefore:

18. Denominators with quadratic factors A similar procedure is applied if one of the factors in the denominator is a quadratic. For example: This results in:

19. Denominators with quadratic factors Equating coefficients of powers of x yields: Three equations in three unknowns with solution:

20. Denominators with repeated factors Repeated factors in the denominator of the original fraction of the form: give partial fractions of the form: Partial fractions

21. Denominators with repeated factors Partial fractions Similarly, repeated factors in the denominator of the original fraction of the form: give partial fractions of the form:

22. Partial fractions Programme F7: Partial fractions Find the partial fractions for the following expressions : 1.7x+18/(x+2)(x+3) 2.2x-7/x²+5x+4 3.-9/2x²+15x+18 4.5x-11/x²-5x+4 5.3x+11/2x²+3x-2

23. Partial fractions Programme F7: Partial fractions Find the partial fractions for the following expressions : 6.x+21/2(2x+3)(3x-2) 7. x-35/x²-25 8.x-4/x²-6x+9 9.5x+4/-x²-x+2 10.12x-5/9x²-6x+1