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© Boardworks Ltd 2005 1 of 73 A1 Algebraic manipulation KS4 Mathematics.

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1 © Boardworks Ltd of 73 A1 Algebraic manipulation KS4 Mathematics

2 © Boardworks Ltd of 73 Contents A A A A A A1.1 Using index laws A1 Algebraic manipulation A1.2 Multiplying out brackets A1.3 Factorization A1.5 Algebraic fractions A1.4 Factorizing quadratic expressions

3 © Boardworks Ltd of 73 Multiplying terms Simplify: x + x + x + x + x = 5 x Simplify: x × x × x × x × xx × x × x × x × x = x 5 x to the power of 5 x 5 as been written using index notation. xnxn The number x is called the base. The number n is called the index or power.

4 © Boardworks Ltd of 73 We can use index notation to simplify expressions. For example, 3 p × 2 p =3 × p × 2 × p =6p26p2 q 2 × q 3 = q × q × q × q × q = q5q5 3 r × r 2 =3 × r × r × r =3r33r3 3 t × 3 t =(3 t ) 2 or9t29t2 Multiplying terms involving indices

5 © Boardworks Ltd of 73 Multiplying terms with the same base For example, a 4 × a 2 =( a × a × a × a ) × ( a × a ) = a × a × a × a × a × a = a 6 When we multiply two terms with the same base the indices are added. = a (4 + 2) In general, x m × x n = x ( m + n )

6 © Boardworks Ltd of 73 Dividing terms Remember, in algebra we do not usually use the division sign, ÷. Instead, we write the number or term we are dividing by underneath like a fraction. For example, ( a + b ) ÷ c is written as a + b c

7 © Boardworks Ltd of 73 Like a fraction, we can often simplify expressions by cancelling. For example, n 3 ÷ n 2 = n3n3 n2n2 = n × n × nn × n × n n × nn × n = n 6 p 2 ÷ 3 p = 6p26p2 3p3p = 6 × p × p 3 × p 2 = 2 p Dividing terms

8 © Boardworks Ltd of 73 Dividing terms with the same base For example, a 5 ÷ a 2 = a × a × a × a × a a × a = a × a × a = a3a3 4 p 6 ÷ 2 p 4 = 4 × p × p × p × p × p × p 2 × p × p × p × p = 2 × p × p = 2p22p2 = a (5 – 2) = 2 p (6 – 4) When we divide two terms with the same base the indices are subtracted. In general, x m ÷ x n = x (m – n) 2

9 © Boardworks Ltd of 73 Hexagon puzzle

10 © Boardworks Ltd of 73 Sometimes terms can be raised to a power and the result raised to another power. For example, ( y 3 ) 2 =( pq 2 ) 4 = Expressions of the form ( x m ) n y 3 × y 3 = ( y × y × y ) × ( y × y × y ) = y 6 pq 2 × pq 2 × pq 2 × pq 2 = p 4 × q ( ) = p 4 × q 8 = p 4 q 8

11 © Boardworks Ltd of 73 Expressions of the form ( x m ) n For example, ( a 5 ) 3 = a 5 × a 5 × a 5 = a ( ) = a 15 When a term is raised to a power and the result raised to another power, the powers are multiplied. = a (3 × 5) In general, ( x m ) n = x mn

12 © Boardworks Ltd of 73 Expressions of the form ( x m ) n Rewrite the following without brackets. 1) (2 a 2 ) 3 =8a68a6 2) ( m 3 n ) 4 = m 12 n 4 3) ( t –4 ) 2 = t–8t–8 4) (3 g 5 ) 3 =27 g 15 5) ( ab –2 ) –2 = a –2 b 4 6) ( p 2 q –5 ) –1 = p–2q5p–2q5 7) ( h ½ ) 2 = h 8) (7 a 4 b –3 ) 0 = 1

13 © Boardworks Ltd of 73 The zero index Look at the following division: y 4 ÷ y 4 =1 But using the rule that x m ÷ x n = x (m – n) y 4 ÷ y 4 = y (4 – 4) = y0y0 That means that y 0 = 1 In general, for all x  0, x0 = 1x0 = 1 x0 = 1x0 = 1 Any number or term divided by itself is equal to 1.

14 © Boardworks Ltd of 73 Negative indices Look at the following division: b 2 ÷ b 4 = b × b b × b × b × b = 1 b × b = 1 b2b2 But using the rule that x m ÷ x n = x (m – n) b 2 ÷ b 4 = b (2 – 4) = b –2 That means that b –2 = 1 b2b2 In general, x–n =x–n = 1 xnxn

15 © Boardworks Ltd of 73 Negative indices Write the following using fraction notation: u –1 = 1 u 2 b –4 = 2 b4b4 x 2 y –3 = x2x2 y3y3 This is the reciprocal of u. 2 a (3 – b) –2 = 2a2a (3 – b) 2

16 © Boardworks Ltd of 73 Negative indices Write the following using negative indices: 2 a = x3x3 y4y4 = p2p2 q + 2 = 3m3m ( n 2 + 2) 3 = 2 a –1 x 3 y –4 p 2 ( q + 2) –1 3 m ( n 2 + 2) –3

17 © Boardworks Ltd of 73 Indices can also be fractional. Fractional indices x × x = x + = x 1 = x But,  x ×  x = x x 1 = x So, x =  x 1 2 Similarly, x × x × x = x + + = But,  x ×  x ×  x = x 333 So, x =  x The square root of x. The cube root of x.

18 © Boardworks Ltd of 73 x =  x In general, Fractional indices Also, we can write x as x. m n 1 n × m× m Using the rule that ( x m ) n = x mn, we can write 1 n n We can also write x as x m ×. m n 1 n x × m = ( x ) m = (  x) m 1 n 1 n n In general, x =  x m x = (  x ) m m n n or m n n x = ( x m ) =  x m 1 n m×m× n 1 n

19 © Boardworks Ltd of 73 Here is a summary of the index laws. x m × x n = x ( m + n ) x m ÷ x n = x (m – n) Index laws ( x m ) n = x mn x1 = xx1 = x x 0 = 1 (for x = 0) x =  x 1 n n 1 2 x =  x m or (  x ) m n m n n x –1 = 1 x x – n = 1 xnxn

20 © Boardworks Ltd of 73 Contents A A A A A A1.2 Multiplying out brackets A1.3 Factorization A1.1 Using index laws A1 Algebraic manipulation A1.5 Algebraic fractions A1.4 Factorizing quadratic expressions

21 © Boardworks Ltd of 73 Look at this algebraic expression: Expanding expressions with brackets 3 y (4 – 2 y ) This means 3 y × (4 – 2 y ), but we do not usually write × in algebra. To expand or multiply out this expression we multiply every term inside the bracket by the term outside the bracket. 3 y (4 – 2 y ) =12 y – 6 y 2

22 © Boardworks Ltd of 73 Look at this algebraic expression: Expanding expressions with brackets – a (2 a 2 – 2 a + 3) When there is a negative term outside the bracket, the signs of the multiplied terms change. – a (2 a 2 – 3 a + 1) =–2 a a 2 – a In general,– x ( y + z ) =– xy – xz – x ( y – z ) =– xy + xz –( y + z ) =– y – z –( y – z ) =– y + z

23 © Boardworks Ltd of 73 Expanding brackets and simplifying Sometimes we need to multiply out brackets and then simplify. For example,3 x + 2 x (5 – x ) We need to multiply the bracket by 2 x and collect together like terms. 3x3x + 10 x – 2 x 2 = 13 x – 2 x 2 3 x + 2 x (5 – x ) =

24 © Boardworks Ltd of 73 Expanding brackets and simplifying Expand and simplify:4 – (5 n – 3) We need to multiply the bracket by –1 and collect together like terms. 4 – 5 n + 3 = – 5 n = 7 – 5 n 4 – (5 n – 3) =

25 © Boardworks Ltd of 73 Expanding brackets and simplifying Expand and simplify:2(3 n – 4) + 3(3 n + 5) We need to multiply out both brackets and collect together like terms. 6n6n – n + 15 = 6 n + 9 n – = 15 n + 7 2(3 n – 4) + 3(3 n + 5) =

26 © Boardworks Ltd of 73 We need to multiply out both brackets and collect together like terms. 15 a + 10 b – 2 a – 5 ab = 15 a – 2 a + 10 b – 5 ab = 13 a + 10 b – 5 ab Expanding brackets then simplifying 5(3 a + 2 b ) – a (2 + 5 b ) = Expand and simplify:5(3 a + 2 b ) – a (2 + 5 b )

27 © Boardworks Ltd of 73 Find the area of the rectangle

28 © Boardworks Ltd of 73 Find the area of the rectangle What is the area of a rectangle of length ( a + b ) and width ( c + d )? a b c d acbc adbd In general, ( a + b )( c + d ) = ac + ad + bc + bd

29 © Boardworks Ltd of 73 Expanding two brackets Look at this algebraic expression: (3 + t )(4 – 2 t ) This means (3 + t ) × (4 – 2 t ), but we do not usually write × in algebra. To expand or multiply out this expression we multiply every term in the second bracket by every term in the first bracket. (3 + t )(4 – 2 t ) =3(4 – 2 t ) + t (4 – 2 t ) =12– 6 t + 4 t – 2 t 2 = 12 – 2 t – 2 t 2 This is a quadratic expression.

30 © Boardworks Ltd of 73 Using the grid method to expand brackets

31 © Boardworks Ltd of 73 Expanding two brackets With practice we can expand the product of two linear expressions in fewer steps. For example, ( x – 5)( x + 2) = x2x2 + 2 x – 5 x – 10 = x 2 – 3 x – 10 Notice that –3 is the sum of –5 and 2 … … and that –10 is the product of –5 and 2.

32 © Boardworks Ltd of 73 Matching quadratic expressions 1

33 © Boardworks Ltd of 73 Matching quadratic expressions 2

34 © Boardworks Ltd of 73 Squaring expressions Expand and simplify: (2 – 3 a ) 2 We can write this as, (2 – 3 a ) 2 = (2 – 3 a )(2 – 3 a ) Expanding, (2 – 3 a )(2 – 3 a ) = 2(2 – 3 a ) – 3 a (2 – 3 a ) =4– 6 a + 9 a 2 = 4 – 12 a + 9 a 2

35 © Boardworks Ltd of 73 Squaring expressions In general, ( a + b ) 2 = a ab + b 2 The first term squared … … plus 2 × the product of the two terms … … plus the second term squared. For example, (3 m + 2 n ) 2 = 9 m mn + 4 n 2

36 © Boardworks Ltd of 73 Squaring expressions

37 © Boardworks Ltd of 73 The difference between two squares Expand and simplify (2 a + 7)(2 a – 7) Expanding, (2 a + 7)(2 a – 7) = 2 a (2 a – 7) + 7(2 a – 7) = 4a24a2 – 14 a + 14 a – 49 = 4 a 2 – 49 When we simplify, the two middle terms cancel out. In general, ( a + b )( a – b ) = a 2 – b 2 This is the difference between two squares.

38 © Boardworks Ltd of 73 The difference between two squares

39 © Boardworks Ltd of 73 Matching the difference between two squares

40 © Boardworks Ltd of 73 Contents A A A A A A1.3 Factorization A1.2 Multiplying out brackets A1.1 Using index laws A1 Algebraic manipulation A1.5 Algebraic fractions A1.4 Factorizing quadratic expressions

41 © Boardworks Ltd of 73 Factorizing expressions Factorizing an expression is the opposite of expanding it. a ( b + c ) ab + ac Expanding or multiplying out Factorizing Often: When we expand an expression we remove the brackets. When we factorize an expression we write it with brackets.

42 © Boardworks Ltd of 73 Factorizing expressions Expressions can be factorized by dividing each term by a common factor and writing this outside of a pair of brackets. For example, in the expression 5 x + 10 the terms 5 x and 10 have a common factor, 5. We can write the 5 outside of a set of brackets 5( x + 2) We can write the 5 outside of a set of brackets and mentally divide 5 x + 10 by 5. (5 x + 10) ÷ 5 = x + 2 This is written inside the bracket. 5( x + 2)

43 © Boardworks Ltd of 73 Factorizing expressions Writing 5 x + 10 as 5( x + 2) is called factorizing the expression. Factorize 6 a a + 8 =2(3 a + 4) Factorize 12 n – 9 n 2 12 n – 9 n 2 =3 n (4 – 3 n ) The highest common factor of 6 a and 8 is 2.2. (6 a + 8) ÷ 2 =3 a + 4 The highest common factor of 12 n and 9 n 2 is 3n.3n. (12 n – 9 n 2 ) ÷ 3 n =4 – 3 n

44 © Boardworks Ltd of 73 Factorizing expressions Writing 5 x + 10 as 5( x + 2) is called factorizing the expression. 3 x + x 2 = x (3 + x ) 2 p + 6 p 2 – 4 p 3 = 2 p (1 + 3 p – 2 p 2 ) The highest common factor of 3 x and x 2 is x.x. (3 x + x 2 ) ÷ x =3 + x The highest common factor of 2 p, 6 p 2 and 4 p 3 is 2p.2p. (2 p + 6 p 2 – 4 p 3 ) ÷ 2 p = p – 2 p 2 Factorize 3 x + x 2 Factorize 2 p + 6 p 2 – 4 p 3

45 © Boardworks Ltd of 73 Factorization

46 © Boardworks Ltd of 73 Factorization by pairing Some expressions containing four terms can be factorized by regrouping the terms into pairs that share a common factor. For example, Factorize 4 a + ab b Two terms share a common factor of 4 and the remaining two terms share a common factor of b. 4 a + ab b = 4 a ab + b = 4( a + 1) + b ( a + 1) 4( a + 1) and + b ( a + 1) share a common factor of ( a + 1) so we can write this as ( a + 1)(4 + b )

47 © Boardworks Ltd of 73 Factorization by pairing Factorize xy – y – 3 x We can regroup the terms in this expression into two pairs of terms that share a common factor. xy – y – 3 x = xy + 2 y – 3 x – 6 = y ( x + 2) – 3( x + 2) y ( x + 2) and – 3( x + 2) share a common factor of ( x + 2) so we can write this as ( x + 2)( y – 3) When we take out a factor of –3, – 6 becomes + 2

48 © Boardworks Ltd of 73 Contents A A A A A A1.4 Factorizing quadratic expressions A1.3 Factorization A1.2 Multiplying out brackets A1.1 Using index laws A1 Algebraic manipulation A1.5 Algebraic fractions

49 © Boardworks Ltd of 73 Quadratic expressions A quadratic expression is an expression in which the highest power of the variable is 2. For example, x 2 – 2, w w + 1,4 – 5 g 2, t2t2 2 The general form of a quadratic expression in x is: x is a variable. a is a fixed number and is the coefficient of x 2. b is a fixed number and is the coefficient of x. c is a fixed number and is a constant term. ax 2 + bx + c (where a = 0)

50 © Boardworks Ltd of 73 Factorizing expressions Remember: factorizing an expression is the opposite of expanding it. Expanding or multiplying out Factorizing Often: When we expand an expression we remove the brackets. ( a + 1)( a + 2) a a + 2 When we factorize an expression we write it with brackets.

51 © Boardworks Ltd of 73 Factorizing quadratic expressions Quadratic expressions of the form x 2 + bx + c can be factorized if they can be written using brackets as ( x + d )( x + e ) where d and e are integers. If we expand ( x + d )( x + e ) we have, ( x + d )( x + e ) = x 2 + dx + ex + de = x 2 + ( d + e ) x + de Comparing this to x 2 + bx + c we can see that: The sum of d and e must be equal to b, the coefficient of x. The product of d and e must be equal to c, the constant term.

52 © Boardworks Ltd of 73 Factorizing quadratic expressions 1

53 © Boardworks Ltd of 73 Matching quadratic expressions 1

54 © Boardworks Ltd of 73 Factorizing quadratic expressions Quadratic expressions of the form ax 2 + bx + c can be factorized if they can be written using brackets as ( dx + e )( fx + g ) where d, e, f and g are integers. If we expand ( dx + e )( fx + g )we have, ( dx + e )( fx + g )= dfx 2 + dgx + efx + eg = dfx 2 + ( dg + ef ) x + eg Comparing this to ax 2 + bx + c we can see that we must choose d, e, f and g such that: a = df, b = ( dg + ef ) c = eg

55 © Boardworks Ltd of 73 Factorizing quadratic expressions 2

56 © Boardworks Ltd of 73 Matching quadratic expressions 2

57 © Boardworks Ltd of 73 Factorizing the difference between two squares A quadratic expression in the form x 2 – a 2 is called the difference between two squares. The difference between two squares can be factorized as follows: x 2 – a 2 = ( x + a )( x – a ) For example, 9 x 2 – 16 = (3 x + 4)(3 x – 4) 25 a 2 – 1 = (5 a + 1)(5 a – 1) m 4 – 49 n 2 = ( m n )( m 2 – 7 n )

58 © Boardworks Ltd of 73 Factorizing the difference between two squares

59 © Boardworks Ltd of 73 Matching the difference between two squares

60 © Boardworks Ltd of 73 Contents A A A A A A1.5 Algebraic fractions A1.3 Factorization A1.2 Multiplying out brackets A1.1 Using index laws A1 Algebraic manipulation A1.4 Factorizing quadratic expressions

61 © Boardworks Ltd of 73 Algebraic fractions The rules that apply to numerical fractions also apply to algebraic fractions. For example, if we multiply or divide the numerator and the denominator of a fraction by the same number or term we produce an equivalent fraction. 3x3x 4x24x2 and are examples of algebraic fractions. 2a2a 3 a + 2 For example, 3x3x 4x24x2 = 3 4x4x = 6 8x8x = 3y3y 4 xy = 3( a + 2) 4 x ( a + 2)

62 © Boardworks Ltd of 73 Equivalent algebraic fractions

63 © Boardworks Ltd of 73 Simplifying algebraic fractions We simplify or cancel algebraic fractions in the same way as numerical fractions, by dividing the numerator and the denominator by common factors. For example, Simplify 6 ab 3 ab 2 6 ab 3 ab 2 = 6 × a × b 3 × a × b × b 2 = 2 b

64 © Boardworks Ltd of 73 Simplifying algebraic fractions Sometimes we need to factorize the numerator and the denominator before we can simplify an algebraic fraction. For example, Simplify 2 a + a a = a 4 2 a + a a = a (2 + a ) 4(2 + a )

65 © Boardworks Ltd of 73 Simplifying algebraic fractions Simplify b 2 – 36 3 b – 18 b 2 – 36 is the difference between two squares. b 2 – 36 3 b – 18 = ( b + 6)( b – 6) 3( b – 6) b = If required, we can write this as 6 3 = b 3 + b 3 + 2

66 © Boardworks Ltd of 73 Manipulating algebraic fractions Remember, a fraction written in the form a + b c can be written as b c a c + However, a fraction written in the form c a + b cannot be written as c b c a + For example, = but =

67 © Boardworks Ltd of 73 Multiplying and dividing algebraic fractions We can multiply and divide algebraic fractions using the same rules that we use for numerical fractions. In general, a b × = c d ac bd a b ÷ = c d a b × = d c ad bc and, For example, 3p3p 4 × = 2 (1 – p ) 6p6p 4(1 – p ) = 3 2 3p3p 2(1 – p )

68 © Boardworks Ltd of y – 6 ÷= 4 y – 2 This is the reciprocal of 4 y – y – 6 × 4 y – 2 2 3( y – 2) ×= 4 y – = Multiplying and dividing algebraic fractions 2 What is 2 3 y – 6 ÷ 4 y – 2 ?

69 © Boardworks Ltd of 73 Adding algebraic fractions We can add algebraic fractions using the same method that we use for numerical fractions. For example, What is 1 a + 2 b ? We need to write the fractions over a common denominator before we can add them. 1 a + 2 b = b + 2 a ab b + 2a2a = In general, += a b c d ad + bc bd

70 © Boardworks Ltd of 73 Adding algebraic fractions What is 3 x + y 2 ? We need to write the fractions over a common denominator before we can add them. 3 x + y 2 == 6 + xy 2x2x + 6 2x2x xy 2x2x = + 3 × 2 x × 2 y × x 2 × x

71 © Boardworks Ltd of 73 Subtracting algebraic fractions We can also subtract algebraic fractions using the same method as we use for numerical fractions. For example, We need to write the fractions over a common denominator before we can subtract them. In general, What is–? p 3 q 2 –= p 3 q 2 –= 2p2p 6 3q3q 6 2 p – 3 q 6 –= a b c d ad – bc bd

72 © Boardworks Ltd of 73 Subtracting algebraic fractions What is–? – (2 + p ) × 2 q 4 × 2 q 3 × 4 2q × 42q × p 4 3 2q2q = – 4 3 2q2q =– 2 q (2 + p ) 8q8q 12 8q8q = 2 q (2 + p ) – 12 8q8q 4 6 = q (2 + p ) – 6 4q4q

73 © Boardworks Ltd of 73 Addition pyramid – algebraic fractions


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