Algebraic Manipulation

Topic: Advanced Algebraic Manipulation Pre-requisite skills: (1) (2) (3) (4)

Expand and Factorise simple algebraic expressions Multiply products of sums of algebraic expressions Multiply & divide algebraic fractions Add & subtract algebraic fractions with linear or quadratic denominators

The sub-topics below are an extension to Sec 2: • • • •

Expand and Factorise more complex algebraic expressions Change the subject of the formula involving square roots Add, Subtract, Multiply and Divide algebraic fractions with quadratic denominators Solve equations involving quadratic expressions

Exercise 1 Expand the following algebraic expressions Q1

16(a + 3)(a − 4) − (2a + 3)(a − 1)

Q2

(2a − 3b) 2 − (b − 3a) 2

Q3

(2 − 3x)(2 + 3x) 2

Q4

(3a − 2)(−5a + a 2 − 1)

Q5

[(2 x + 3 y ) − 3(3x − y )]2

Q6

2(3x − 4 y ) 2 − (7 x + 5 y ) 2

Q7

(a + b − c)2 − (a − b + c) 2

Q8

(a + 1) 2 − 3(a 2 − a + 2) + (a − 2) 2

Q9

(3a + 2)(b − 1) − (2ab + 1) + (a − b) 2

Q10

2 x( x − 2)( x + 3) − (−2 x 2 + x)( x − 1)

Derivation of Pascal’s Triangle

Q1

( a + b) 2

Q2

( a + b)3 [Hint: (a + b)3 = (a + b)(a + b) 2 ]

1

( a + b) 4 [Hint: (a + b) 4 = (a + b)(a + b)3 ]

Q3

Observe the solutions for Q1 – 3 and derive the Pascal’s Triangle:

Use the Pascal’s Triangle to expand (a + b)5 . How would you modify the use of Pascal’s Triangle to expand (a − b)3 , (a − b) 4 , (a − b)5 ? Q4

( a + b)5

Q5

( a − b)3

Q6

( a − b) 4

Q7

( a − b)5

2

Exercise 2 Factorise the following algebraic expressions

Q1

1 + 3(2 x + 1) + 2(2 x + 1) 2

Q2

(a − 2b) 2 + (a + b) 2 + 2(a − 2b)(a + b)

Q3

( x + 2) 2 − 2( x 2 + x − 2) + ( x − 1) 2

Q4

4(3a − 1) 2 − 2(3a − 1) − 2

Q5

(b − 4) 2 + 4 + 4(4 − b)

Q6

2 y + 1 + 2(2 y + 1) 2 + (2 y + 1)3

Q7

a3 − a

Q8

m + (m 2 − m − 12)a 2 + (2m 2 − 4)a + m 2

Q9

50 xy 2 − 72 x 3 z 2

Q10

4 x 2 + 4 xy + y 2 − z 2

Q11

x( x + 1) − y ( y + 1)

Q12

x3 + 3x 2 − 4

Q13

49 x 2 y 2 + 44 xy + 9 − x 2 − y 2

Q14

a 4 + 2a 3 + 3a 2 + 2a + 1

Q15

1 − 2 x + 3x 2 − 2 x3 + x 4

Change the subject of the formula

Eg 1: Einstein’s Formula E = mc 2 , where E is energy, m is mass and c is the speed of light

Eg 2: Power Formula P = I 2 R , where P is power, I is current and R is electrical resistance

1 Eg 3: Distance Formula s = ut + at 2 , where s is distance, u is initial velocity, t is time and a is 2 acceleration

Exercise 3

Changing the subject of the formula Shinglee New Syllabus Mathematics 3 Pg 34 Ex. 2F Q2 3

Multiply and Divide Algebraic Fractions

Eg 1:

2x2 + x − 3 x2 + 7 x − 8 x2 + 8x ÷ × x 2 + x − 6 x 2 − 4 x + 4 −2 x 2 + x + 6

Eg 2:

x y − y x 1 1 + x y

Eg 3:

a −b b−c c−a × × a −c b−a c−b

Exercise 4

Multiply and Divide Algebraic Fractions Shinglee New Syllabus Mathematics 3 Pg 36 Ex. 2G

4

Add and Subtract Algebraic Fractions

Eg 1:

3y 8x + x − 2 y 2 y − 4x

Eg 2:

3x 2x −1 − x − 2 x +1

Eg 3:

1 2 5 − + 2x − 3 3 − 2x 9 − 4x2

Exercise 5

Add and Subtract Algebraic Fractions Shinglee New Syllabus Mathematics 3 Pg 37 Ex. 2H

5

Equations involving Algebraic Fractions

Eg 1:

4 5 91 − = 2 3 − x x −1 x − 4x + 3

Eg 2:

6 7 − =0 2 x − 1 3x + 2

Eg 3:

m+3 m−3 1 + 2 = 2 m + m m −1 2

Exercise 6

Equations involving Algebraic Fractions Shinglee New Syllabus Mathematics 3 Pg 39 Ex. 2I

6