Chapter 2_ Factorization & Algebraic Fractions F2.pptx

Maths

1. Chapter 2 FACTORIZATION & ALGEBRAIC FRACTIONS F

2. Expansion is the process of expressing a product of algebraic expressions as a sum or difference of expressions. For instance, Expanding Brackets

3. When we expand single brackets, we multiply each term in the brackets by the term outside the brackets. Example 1: Expand each of the following. (a) q(r + s) (b) 3q (4r - 6s) (c) -5k(2m +  3n)  Solution: (a) (b ) (c)

4. To expand two brackets, we multiply each term in the first brackets by each of the terms in the second brackets, then add and subtract the products. For instance, 2. - (a + b)(c + d) = ac + ad + bc + bd     - (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2     - (a + b) 2 = (a - b)(a - b) = a 2 - 2ab + b 2     - ( a + b)(a - b) = a 2 - b 2 Example 2: Expand each of the following. (a) (q + 2)(q + 5) (b) (m + 3)(m - 3) (c) (2n - 4) 2

5. Solution: (a) (b) (c)

6. 1. An algebraic term is made up of a numerical coefficient and a variable. Example 2. Factors of an algebraic term can be multiplied together to produce the original term. Example: The factors of 4a are 4, a  4a and 1. Factorization of Algebraic Expressions Example 3: State the factors of 5mn. Solution: The factors of 5mn are: 1, 5mn, 5, m, n, 5m, 5p, mn

7. 1. Common factors are factors shared by two or more terms Examples: (a) Factors of pq are 1, pq, p, q. (b) Factors of pr are 1 , pr, p, r.        The common factors of pq and pr are 1. 2. The highest Common Factor (HCF) is the common factor which has the largest value. Examples (a) Factors of 6q are 1, 2, 3, 6, q, 2q, 3q, 6q. (b) Factors of 3q are 1, 3, q, 3q.       The common factors of 6q and 3q are 1, 3, q and 3q.

8. Example 4: State teh common factors and the HCF of 12a and 14a. Solution: Factors of 12a: 1, 2, 3,4, 5, 6, 12, a, 2a, 3a, 4a, 6a, 12a. Factors of  14a: 1, 2, 7, 14, a, 2a, 7a, 14a. (a) The common factors are 1, 2, a, 2a. (b) The HCF is 2a.

9. 1. Factorizing an algebraic expressions means selecting the HCF of the terms, then inserting  brackets to make it a product. That is, to factorize is to change the expression (which     involves a sum or difference) into a product. Factorization is the reverse of expansion. Example: Example 5: Factories the following. (a) 8 x - 12 (b) pq + pr + qs + rs Solution: (a) 6 b - 15      = 3 x 2 x b  - 3 x 5      = 3(2 x - 5) (b) pq + pr + qs + rs       = p(q + r) + s(q + r)       = (q + r)(p + s)

10. 2. Factorizing is also done using the difference of two squares. Example: 3. Factorizing and simplifying algebraic fractions are done as follows: (a) Factories the numerator and the denominator separately. (b) Cancel the common factors in the numerator and denominator. Example:

11. Example 6: Factorize and simplify the following. (a)  (b) Solution: (a) (b)

12. An algebraic fraction has the form: Examples Addition and Subtraction of Algebraic Fractions

13. To add or subtract two algebraic  fractions with the same denominator, we add or subtract the numerators and retain the common denominator. Example 7: Simplify  the following. Solution:

14. To add or subtract two algebraic fractions with one denominator as a multiple of the other denominator, follow the following steps: Find the LCM of the denominators. 2. Change each of the algebraic fractions to an equivalent fraction with the LCM as the denominator. 3. Add or subtract the numerators and retain the common denominator. 4. Simplify wherever possible.

15. Example 8: Solution:

16. 1. To add subtract two algebraic fractions with denominators that have no common factor except 1, follow the steps in learning outcome H above. The LCM of the denominators is the product of the two denominators. Example 9: Simplify the following. Solution:

17. 2. Similarly, to add or subtract two algebraic fractions with denominators that have a  common factor, follow the steps in Learning Outcome H above. Example 10: Simplify the following. Solution:

18. Multiplication and Division of Algebraic Fractions To multiply algebraic fractions, multiply the numerator by the numerator and the denominator by the denominator. Example 11: Simplify the following (a) (b)

19. Solution: ( c ) (d) Question : ( c ) (d)

20. To divide algebraic fractions: - Multiply by the reciprocal of the divisor. - Cancel common factors, if there's any. - Multiply the numerator by the numerator and the denominator by the denominator. Example 12: (a) (b) (c) Solution: (a)

21. To divide an algebraic expression by an algebraic fraction, multiply the algebraic expression by the reciprocal of the fraction. Example 13: Simplify  Solution:

22. To divide an algebraic fraction by an algebraic expression, multiply the algebraic fraction by the reciprocal of the algebraic expression. Example 14: Simplify the following. Solution:

23. Division of two algebraic fractions involves factorization and simplification using common factors and the difference of two squares. Example 15: Simplify: Solution:

24. Common Errors