Algebraic Techniques

Algebraic Techniques

1. 1.01: Index Laws

   • a^M * a^N = a^m+n
   • a^m + a^n = a ^ m-n
   • (a^m)^n = a^mn
   • (ab)^n = a^n * b^n
   • (a/b)^n = (a^n)/(b^n)

2. 1.02: Zero and Negative Indicies

   • x^0 = 1
   • x^-n = 1/x^n

3. 1.03: Fractional Indicies

   • Fractional Indices turn the power into the root of that number:
     • For Example; 4^1/2 = square root of 4
   • Power of 1/n
     • a ^1/n = n root a
   • further fractional indices
     • a ^-1/n = 1/ n root a
     • a ^ m/n = n root (a^m)
     • (a/b)^-n = (b/a)^n

4. 1.04: Simplifying Algebraic Expressions

   • Group like terms
     • For Example; 4x^2 - 3x^2 + 6x^2 = x^2 + 6x^2 = 7x^2

5. 1.05: Expansion

   • Expanding involves removing the brackets from an equation
     • a(b+c) = ab + ac

6. 1.06: Binomial Products

   • Binomial expressions consist of 2 terms. Binomial products are two terms multiplied by each other
     • (x+a)(x+b) = x^2 + bx + ax + ab

7. 1.07: Special Products

   • Difference of two squares
     • (a+b)(a-b) = a^2 - b^2
   • Perfect Squares
     • (a+b)^2 = a^2 + 2ab + b^2
     • (a-b)^2 = a^2 - 2ab + b^2
   • Difference of two squares
     • (a+b)(a-b) = a^2 - b^2
   • Perfect squares
     • (a+b)^2 = a^2 + 2ab + b^2
     • (a-b)^2 = a^2 - 2ab + b^2

8. 1.08: Factorisation

   • To factorise an expression, we use the distributive law in the opposite way from when we expand brackets
     • ax + bx = x (a+b)
   • To factorise an expression, we use the distributive law in the opposite way from when we expand brackets
     • ax + bx = x(a + b)

9. 1.09 Factorising by grouping in pairs

   • If an expression has 4 terms, it can sometimes be factorised in pairs.
   • If an expression has 4 terms, it can sometimes be factorised in pairs.
     • ax + bx + ay + by = x(a+b) + y(a+b) = (a+b)(x+y)

10. 1.10 Factorising Trinomials

    • FInd the values for a and b so that the sum a + b is the middle term and the product ab is the last term
      • x^2 + (a+b)x + ab = (x+a) (x+b)

11. 1.11 Further trinomials

    • When the coefficient of the first term is not 1, for example 5x^2 - 13x + 6 , we need to use a different method to factorise the trinomial
      • This Method still involves finding 2 numbers that five a required sum and product but it also involves grouping in pairs.
        • For example, 5x^2 - 13x +6 = 5x^2 - 10x - 3x + 6 = 5x(x-2) - 3(x-2) = (x-2)(5x-3)
          • First, multiply the coefficient of the first term by the last term; 5 X 6 = 30. Now a + b = -13 and ab = 30. Since the sum is negative and the product is positive, a and b must both be negative. 2 numbers with the product 30 and sum -13 are - 10 and -3. Now write the trinomial with the middle term split into 2 terms, -10x and -3x, and then factorise by grouping in pairs

12. 1.12 perfect squares

    • You have to look at expanding (a+b)^2 = a^2 + 2ab + b^2 and (a-b)^2 = a^2 - 2ab + b^2. These are perfect squares. When factorising, use these results the other way around

13. 1.13 Difference of two squares

14. 1.14 Mixed factorisation

15. 1.15 simplifying algebraic fractions

16. 1.16 Operations with Algebraic fractions

17. 1.17 Substitution

18. 1.18 Simplifying surds

19. 1.19 operations with surds

20. 1.20 rationalising the denominator