1. Business Mathematics Week 12 BOOK C, Unit 9 Expanding Algebra MU123

2. Learning Outcomes After studying this unit, student should be able to: find the sum of any arithmetic sequence prove simple number patterns involving square numbers multiply out pairs of brackets add, subtract, multiply and divide algebraic fractions Week 12

3. We learned that to multiply out the brackets: Multiply each term inside the brackets by the multiplier of the bracket. Solve these two examples: 3m(−2n + 3r−6), (n-1)n Solution: = −6mn + 9mr − 18m, n 2 -n Now, You can multiply out two brackets of the form (a + b)(c + d) in two steps, as follows: First, keep (a + b) as one expression and use it as the multiplier to expand the right bracket (c + d), to obtain (a + b)(c + d) = (a+b)c + (a+b)d. Second, expand each of the (a + b) brackets on the right-hand side: (a + b)(c + d) = (a+b)c + (a+b)d = ac + bc + ad + bd. 2 Multiplying out pairs of brackets 2.1 Pairs of brackets

4. We have done the following: The acronym FOIL may help you to remember the order in which these pairs are multiplied: FOIL stands for: (1) First: ac, (2) Outer: ad, (3) Inner: bc, (4) Last: bd. 2 Multiplying out pairs of brackets 2.1 Pairs of brackets

5. For example, you can use the strategy above to multiply out the product (2s − t)(u − 3v): Apply FOIL: F:2s × u = 2su, O:2s × (−3v) = −6sv, I:(−t) × u = −tu, L:(−t) × (−3v) = 3tv. Then: 2su − 6sv − tu + 3tv Exercise: multiply out the brackets: (a + 2b)(3c − d) Solution:3ac - ad + 6bc – 2bd 2 Multiplying out pairs of brackets 2.1 Pairs of brackets

6. Exercise: Multiply the following brackets: (n − 2)(n + 2) Solution: n 2 + 2n -2n -4 = n 2 – 4 (x – 3) 2 Solution: (X – 3)(X – 3) = X 2 – 3X -3X + 9= X 2 -6X + 9 We can also use the first strategy to multiply out two brackets that contain more than two terms: Multiply the following brackets : (2a − b)(c − 3d + 2e) Solution: = 2a(c−3d + 2e)−b(c−3d + 2e) = 2ac − 6ad + 4ae − bc + 3bd − 2be 2 Multiplying out pairs of brackets 2.1 Pairs of brackets

7. 4 Manipulating algebraic fractions 4.1 Equivalent algebraic fractions

8. 4 Manipulating algebraic fractions 4.2 Adding and subtracting algebraic fractions

9. 4 Manipulating algebraic fractions 4.2 Adding and subtracting algebraic fractions

10. 4 Manipulating algebraic fractions 4.2 Adding and subtracting algebraic fractions

11. 4 Manipulating algebraic fractions 4.3 Multiplying and dividing algebraic fractions

12. 4 Manipulating algebraic fractions 4.3 Multiplying and dividing algebraic fractions

13. Thank you