1)-Algebraic-Expressions.pptx IAL P1 CHAPTER 1 PPT

IAL PPT P1

1. Algebraic Expressions Twitter: @Owen134866 www.mathsfreeresourcelibrary.com

2. Prior Knowledge Check Simplify: a) b) 2) Write as a single power of 2 a) b) c) 3) Expand: a) b) c)   4) Write down the highest common factor of: 24 and 16 b) and c) and 5) Simplify: a) b) c)                              

3. Teachings for Exercise 1A

4. Algebraic Expressions You can use the laws of indices to simplify powers of the same base   1A a)     b)     c)     d)     e)       f)      

5. Algebraic Expressions You can use the laws of indices to simplify powers of the same base   1A Expand and simplify if possible   a)     b)     c)     d)     LO: To be able to apply laws of indices to simplify expressions:

6. Algebraic Expressions You can use the laws of indices to simplify powers of the same base   1A Simplify   a)     b)           c)       If you have a single term as the denominator, you can simplify the numerator terms separately…

7. Teachings for Exercise 1B

8. Algebraic Expressions To find the product of two expressions you multiply each term in one expression by each term in the other expression 1B (x + 4)(x + 7)  x 2 + 4x + 7x + 28  x 2 + 11x + 28 (2x + 3)(x – 8)  2x 2 + 3x – 16x – 24  2x 2 – 13x - 24 + 28 + 7x + 7 + 4x x 2 x + 4 x - 24 - 16x - 8 + 3x 2x 2 x + 3 2x There are various methods for doing this, all are ok!

9. Algebraic Expressions To find the product of two expressions you multiply each term in one expression by each term in the other expression If you have more than two brackets, just multiply any 2 first, and then multiply the answer by the next one 1B Expand                 Multiply the first pair of brackets Multiply this new pair Simplify

10. Teachings for Exercise 1C

11. Algebraic Expressions You can write expressions as products of their factors. This is known as factorising . If the terms have a common factor (or several), then the expression can be factorized into a single bracket 1C a) Common Factor 3 b) x c) 4x d) 3xy e) 3x

12. x 2 + 3x 2 + You get the last number in a Quadratic Equation by multiplying the 2 numbers in the brackets You get the middle number by adding the 2 numbers in the brackets (x + 2)(x + 1) Algebraic Expressions

13. x 2 - 2x 15 - You get the last number in a Quadratic Equation by multiplying the 2 numbers in the brackets You get the middle number by adding the 2 numbers in the brackets (x - 5)(x + 3) Algebraic Expressions

14. x 2 - 7x + 12 Numbers that multiply to give + 12 +3 +4 -3 -4 +12 +1 -12 -1 +6 +2 -6 -2 Which pair adds to give -7? (x - 3)(x - 4) So the brackets were originally… Algebraic Expressions

15. x 2 + 10x + 16 Numbers that multiply to give + 16 +1 +16 -1 -16 +2 +8 -2 -8 +4 +4 -4 -4 Which pair adds to give +10? (x + 2)(x + 8) So the brackets were originally… Algebraic Expressions

16. x 2 - x - 20 Numbers that multiply to give - 20 +1 -20 -1 +20 +2 -10 -2 +10 +4 -5 -4 +5 Which pair adds to give - 1? (x + 4)(x - 5) So the brackets were originally… Algebraic Expressions

17. Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. REMEMBER  An equation with an ‘x 2 ’ in does not necessarily go into 2 brackets. You use 2 brackets when there are NO ‘Common Factors’ 1E Examples a) The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’ Algebraic Expressions

18. Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. 1E Examples b) The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’ Algebraic Expressions

19. Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. 1E Examples c) The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’ (In this case, b = 0) This is known as ‘the difference of two squares’  x 2 – y 2 = (x + y)(x – y) Algebraic Expressions

20. Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. 1E Examples d) The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’ Algebraic Expressions

21. Factorising Quadratics A Quadratic Equation has the form; ax 2 + bx + c Where a, b and c are constants and a ≠ 0. You can also Factorise these equations. 1E Examples d) The 2 numbers in brackets must: Multiply to give ‘c’ Add to give ‘b’ Sometimes, you need to remove a ‘common factor’ first… Algebraic Expressions

22. Expand the following pairs of brackets (x + 3)(x + 4)  x 2 + 3x + 4x + 12  x 2 + 7x + 12 (2x + 3)(x + 4)  2x 2 + 3x + 8x + 12  2x 2 + 11x + 12 + 12 + 4x + 4 + 3x x 2 x + 3 x + 12 + 8x + 4 + 3x 2x 2 x + 3 2x When an x term has a ‘2’ coefficient, the rules are different… 2 of the terms are doubled  So, the numbers in the brackets add to give the x term, WHEN ONE HAS BEEN DOUBLED FIRST Algebraic Expressions

23. 2x 2 - 5x - 3 Numbers that multiply to give - 3 -3 +1 +3 -1 One of the values to the left will be doubled when the brackets are expanded (2x + 1)(x - 3) So the brackets were originally… -6 +1 -3 +2 +6 -1 +3 -2 The -3 doubles so it must be on the opposite side to the ‘2x’ Algebraic Expressions

24. 2x 2 + 13x + 11 Numbers that multiply to give + 11 +11 +1 -11 -1 One of the values to the left will be doubled when the brackets are expanded (2x + 11)(x + 1) So the brackets were originally… +22 +1 +11 +2 -22 -1 -11 -2 The +1 doubles so it must be on the opposite side to the ‘2x’ Algebraic Expressions

25. 3x 2 - 11x - 4 Numbers that multiply to give - 4 +2 -2 -4 +1 +4 -1 One of the values to the left will be tripled when the brackets are expanded (3x + 1)(x - 4) So the brackets were originally… +6 -2 +2 -6 -12 +1 -4 +3 The -4 triples so it must be on the opposite side to the ‘3x’ +12 -1 +4 -3 Algebraic Expressions

26. Teachings for Exercise 1D

27. Algebraic Expressions Indices can be negative numbers or fractions   1D Simplify a)     b)     c)     d)        

28. Algebraic Expressions Indices can be negative numbers or fractions   1D Simplify e)         Either of these forms is correct – check if the question asks for a specific one! Simplify separately Rewrite

29. Algebraic Expressions Indices can be negative numbers or fractions   1D Evaluate (work out the value of) a)       b)       c)       d)         You can use a calculator for these, but you still need to be able to show the process, especially for algebraic versions

30. Algebraic Expressions Indices can be negative numbers or fractions   1D Given that , express in the form where and are constants           Rewrite based on the question Each part is raised to a power ½ Simplify

31. Algebraic Expressions Indices can be negative numbers or fractions   1D Given that , express in the form where and are constants           Rewrite based on the question Each part is raised to a power -1, and will then be multiplied by 4 Simplify   Simplify more

32. Teachings for Exercise 1E

33. Algebraic Expressions In is an integer that is not a square number, then is a surd. It is an example of an irrational number. S urds can be used to leave answers exact without rounding errors, and can be manipulated by using the following rules:   1E Simplify a)       Make sure that what you write is clear…  and are different!   b)         Find a factor which is a square number, which you can then square root Simplify the numerator Simplify the whole fraction

34. Algebraic Expressions In is an integer that is not a square number, then is a surd. It is an example of an irrational number. S urds can be used to leave answers exact without rounding errors, and can be manipulated by using the following rules:   1E Simplify c)   Try to find a common factor       Square roots can be worked out Simplify

35. Algebraic Expressions In is an integer that is not a square number, then is a surd. It is an example of an irrational number. S urds can be used to leave answers exact without rounding errors, and can be manipulated by using the following rules:   1E Expand and simplify if possible a)   Multiply out   b)   Multiply out       Group together like terms. Calculate root 9 Simplify

36. Teachings for Exercise 1F

37. Algebraic Expressions If a fraction has a surd in the denominator, then it can be useful to rearrange it so that the denominator is a rational number. This is called rationalising the denominator. For fractions of the form , multiply the numerator and denominator by For fractions of the form , multiply the numerator and denominator by For fractions of the form , multiply the numerator and denominator by   1F Rationalise a)         b)             Multiply so that the surd is removed from the denominator Multiply both numerator and denominator Multiply out the brackets Simplify

38. Algebraic Expressions If a fraction has a surd in the denominator, then it can be useful to rearrange it so that the denominator is a rational number. This is called rationalising the denominator. For fractions of the form , multiply the numerator and denominator by For fractions of the form , multiply the numerator and denominator by For fractions of the form , multiply the numerator and denominator by   1F Rationalise c)             Multiply both numerator and denominator Multiply out the brackets Simplify

39. Algebraic Expressions If a fraction has a surd in the denominator, then it can be useful to rearrange it so that the denominator is a rational number. This is called rationalising the denominator. For fractions of the form , multiply the numerator and denominator by For fractions of the form , multiply the numerator and denominator by For fractions of the form , multiply the numerator and denominator by   1F Rationalise d)   Multiply out the brackets first                 Multiply to cancel the surds Multiply out the brackets Simplify Divide all by 2