Form 3 Mathematics

Chapter 2: Algebra 1

2.1. Introduction
2.2. Lowest common multiple (LCM) and highest common factor (HCF)
2.2.1. Lowest common multiple (LCM)
2.2.2. Highest common factor (HCF)
2.3. Algebraic manipulations
2.3.1. Linear expressions
2.3.1.1. Expressions with brackets
2.3.1.2. Expressions with fractions
2.3.2. Factorisation
2.3.2.1. Factorising two terms
2.3.2.2. Factorising four terms
2.3.2.3. Factorising three terms (quadratic expressions)
2.3.3. Substitution
2.4. Algebraic fractions
2.4.1. Lowest terms
2.4.2. Addition and subtraction of algebraic fractions
2.4.3. Multiplication and division of algebraic fractions
2.5. Summary
2.6. Further reading
2.7. Test 2

2.1. Introduction

This chapter introduces the topic on Algebra which is part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations. This chapter seeks to equip you with the skills required in dealing with manipulation of algebraic expressions. Algebraic manipulation is one of the most basic, necessary and important skills which should be in a problem solver’s stock, as without it a problem solver would hopelessly be stuck on innumerable problems. The skill of algebraic manipulation is acquired through practice and solving problems.

Objectives

After going through this chapter, you should able to

  • manipulate algebraic processes.
  • find the lowest common multiple (LCM) of algebraic expressions.
  • find the highest common factor (HCF) of algebraic expressions.
  • simplify algebraic fractions.

Key terms

Algebra – Use of letters of the alphabet to stand for numbers.
Algebraic expression – A mathematical statement were letters and numbers are used together for example 2x, 4a + 5, 0.5y + 6z.
Expand – Remove brackets from an expression by multiplication.
Factorise – Writing an expression in terms of its factors.
Quadractic expression – An expression where the highest power of the unknown is 2.
Substitute – To replace a letter with a number.

Time

You should not spend more than 10 hours in this chapter.

Study skills

In order to understand problems in this chapter, you should be able to deal with directed numbers. The key to mastery of mathematics is practice. You need to work out as many problems as possible about algebra to have a better understanding of the subject.

2.2. Lowest common multiple (LCM) and highest common factor (HCF)

2.2.1. Lowest common multiple (LCM)

Lowest common multiple is the smallest number that can be divided by the given numbers without leaving a remainder.

Given the numbers 12 and 18,you can find the lowest common multiple by first listing the multiples of 12 and 18 and identifying the smallest whole number which is divisible by both numbers.

  • Multiples of 12 = 12, 24, 36, 48
  • Multiples of 18 = 18, 36, 54, 72
  • LCM = 36

You can also find the LCM through the use of prime factors. To find the LCM of 12 and 18 you start by writing each number as a product of its prime factors.

  • 12 = 2 × 2 × 3
  • 18 = 2 × 3 × 3
  • Take every factor where it is appearing the highest number of times.
  • LCM = 2 × 2 × 3 × 3
  • LCM = 36

The same method can be used when dealing with algebraic expressions. For example you may be asked to find the LCM of \( 12a^2b \) and \( 18ab^2 \). This is done by splitting the two into their respective prime factors.

  • \( 12a^2b = 2 \times 2 \times 3 \times a \times a \times b \)
  • \( 18ab^2 = 2 \times 3 \times 3 \times a \times b \times b \)
  • \( LCM = 2 \times 2 \times 3 \times 3 \times a \times a \times b \times b \)
  • \( LCM = 36a^2b^2 \)

2.2.2. Highest common factor (HCF)

The highest common factor (HCF) of two whole numbers is the largest whole number which is a factor of both. To find HCF of 12 and 18 you list the factors of 12 and 18.

  • Factors of 12 = 1, 2, 3, 4, 6, 12
  • Factors of 18 = 1, 2, 3, 6, 9, 18
  • Common factors (CF) = 1, 2, 3, 6
  • HCF = 6

Alternatively, the HCF can be found by using prime factors.

  • 12 = 2 × 2 × 3
  • 18 = 2 × 3 × 3
  • HCF = 2 × 3
  • HCF = 6

Example 1
Questions

  1. Find the HCF of \( 18ab^2 \) and \( 24a^2b \)

Answers

  1. \( 18ab^2 = 2 \times 3 \times 3 \times a \times b \times b \)
    \( 24a^2b = 2 \times 2 \times 2 \times 3 \times a \times a \times b \)
    CF = 2 × 3 × a × b
    HCF = 6ab

2.3. Algebraic manipulations

Have you ever encountered an algebraic expression? If yes, list any two algebraic expressions in the space provided.

Here are some examples of algebraic expressions, 3x + y, 4z − 5e and 5p − 3r + 4w.

2.3.1. Linear expressions

Linear expressions are algebraic expressions that contain a variable, and the highest power of the unknown is 1. For example in the expression 2x + 4, x is the variable and the highest power of x is 1.When simplifying linear expressions we collect like terms.

Example 2
Questions

  • Simplify the following expression, 3x + 4y − 8x + 7y + 9.

Answers

  • 3x + 4y − 8x + 7y + 9 (collect like terms)
    3x − 8x + 4y + 7y + 9 (simplify)
    −5x + 11y + 9

2.3.1.1. Expressions with brackets

When removing brackets, the term outside the brackets multiplies each term inside the brackets. For example,

  • a(b + c) = ab + ac
  • –a(b + c) = –ab – ac
  • (a + b)(m + n) = am + an + bm + bn
Note: If the term outside the brackets has a negative sign, the signs inside the brackets change.

Example 3
Questions

  • Simplify the following expressions.
    1. 2x(3y − 4x + 5)
    2. 3x(2 + m) − 4(3 − y)
    3. 6 − 4(x + y)
    4. (3x + y)(2a + b)
    5. \( -2x^2(x + 3y - \dfrac{1}{x}) \)
    6. \( \dfrac{-2}{x}(-x + 4y + \dfrac{1}{x}) \)

Answers

  1. \( 2x(3y − 4x + 5) \\ = 6xy - 8x^2 + 10x \)
  2. 3x(2 + m) − 4(3 − y)
    = 6x + 3mx − 12 + 4y
  3. 6 − 4(x + y)
    = 6 − 4x − 4y
  4. (3x + y)(2a + b)
    = 3x(2a + b) + y(2a + b)
    = 6ax + 3bx + 2ay + by
  5. \( -2x^2(x + 3y - \dfrac{1}{x}) \\ = 2x^3 - 6x^2y + 2x \)
  6. \( \dfrac{-2}{x}(-x + 4y + \dfrac{1}{x}) \\ = \dfrac{2x}{x} - \dfrac{8y}{x} - \dfrac{2}{x^2} \\ = 2 - \dfrac{8y}{x} - \dfrac{2}{x^2} \)

2.3.1.2. Expressions with fractions

Just like the ordinary fractions you dealt with in chapter 1, expressions with fractions can be added or subtracted.

Example 4
Questions

  • Simplify the following fractions.
    1. \( \dfrac{3}{a} + \dfrac{4}{b} + \dfrac{5}{c} \)
    2. \( \dfrac{2x + 3}{5} + \dfrac{x + 1}{4} \)

Answers

  1. \( \dfrac{3}{a} + \dfrac{4}{b} + \dfrac{5}{c} \\ = \dfrac{3bc + 4ac + 5ab}{abc} \)
  2. \( \dfrac{2x + 3}{5} + \dfrac{x + 1}{4} \\ = \dfrac{4(2x +3) + 5(x + 1)}{20} \\ = \dfrac{8x + 12 + 5x + 5}{20} \\ = \dfrac{13x + 17}{20} \)

You may attempt the following exercise.

Exercise 2.1: Algebraic manipulations
Questions

  1. Simplify the following.
    1. a – 3b + 4c – 10a + 5b
    2. 4(m + 3n − p)
    3. 2(x + 3y) + 4(x − 5y)
    4. 2a(a + 3) + 3(2a + 4)
    5. (3x − y)(2x − 3y)
  2. Express the following as single fractions.
    1. \( \dfrac{2a + 3}{4} + \dfrac{a - 1}{4} \)
    2. \( \dfrac{x - 2}{3} - \dfrac{2x + 3}{5} \)
    3. \( \dfrac{m - 3}{4b} - \dfrac{m - 2}{8b} + 5 \)
    3. Hint: The common denominator should be lowest common multiple of the denominators.

Answers

    1. −9a + 2b + 4
    2. 4m + 12n − 4p
    3. 6x − 14y
    4. \( 2a^2 + 12a + 12 \)
    5. \( 6x^2 − 11xy + 3y^2 \)
    1. \( \dfrac{3a + 2}{4} \)
    2. \( \dfrac{-x - 19}{15} \)
    3. \( \dfrac{m - 4 + 40b}{8b} \)

2.3.2. Factorisation

An expression can be written in terms of its factors. This process is called factorisation. To factorise, for example, the expression 6b + 9ab, you write the HCF outside the bracket and divide each term inside the bracket by the HCF.

  • \( 6b + 9ab \\ = 3b(\dfrac{6b}{3b} + \dfrac{9ab}{3b}) \\ = 3b(2 + 3a) \)

Whenever you are factorise, first identify the HCF that you are going to write outside the brackets and then divide each term by the HCF.

2.3.2.1. Factorising two terms

When factorising two terms, identify the HCF and write it outside the brackets.

Example 5
Questions

  • Factorise 8ab + 12a

Answers

  • \( 8ab + 12a \\ = 4a(\dfrac{8ab}{4a} + \dfrac{12a}{4a}) \\ = 4a(2b + 3) \)

2.3.2.2. Factorising four terms

When factorising four terms, group the terms in pairs and then factorise.

Example 6
Questions

  • Factorise.
    1. ax + ay + bx + by
    2. \( 6a^2 − 3a + 4a − 2 \)
    3. 6mn + bd − 3bm − 2nd

Answers

  1. ax + ay + bx + by
    = a(x + y) + b(x + y)
    = (x + y)(a + b)
  2. \( 6a^2 − 3a + 4a − 2 \\ = 3a(2a − 1) + 2(2a − 1) \\ = (2a − 1)(3a + 2) \)
  3. 6mn + bd − 3bm − 2nd. Rearrange equation because the first two terms have no common factor.
    = 6mn − 3bm − 2nd + bd
    = 3m(2n − b) − d(2n − b)
    = (2n − b)(3m − d)

2.3.2.3. Factorising three terms (quadratic expressions)
Using the Product, Sum and Factor method

Expressions of the form \( ax^2 + bx + c \) (quadratic expressions) are factorised using the Product, Sum and Factor method (P.S.F).

Example 7
Questions

  • Factorise the following expressions.
    1. \( x^2 + 7x + 12 \)
    2. \( 3x^2 − 4x + 1 \)
    3. \( a^2 − a − 20 \)

Answers

  1. \( x^2 + 7x + 12 \)
    \( P = x^2 \times 12 = 12x^2 \)
    \( S = 7x \)
    \( F = 4x + 3x \)
    The factors (F) should give a product (P) and sum (S).
    Replace 7x with 4x + 3x so that the terms become 4.
    \( = x^2 + 4x + 3x + 12 \\ = x(x + 4) + 3(x + 4) \\ = (x + 3)(x + 4) \)
  2. \( 3x^2 - 4x + 1 \)
    \( P = 3x^2 \times 1 = 3x^2 \)
    \( S = -4x \)
    \( F = -3x - x \)
    The factors (F) should give a product (P) and sum (S).
    Replace -4x with -3x - x so that the terms become 4.
    \( = 3x^2 - 3x - x + 1 \\ = 3x(x - 1) - 1(x - 1) \\ = (3x - 1)(x - 1) \)
  3. \( a^2 - a - 20 \)
    \( P = a^2 \times -20 = -20a^2 \)
    \( S = -a \)
    \( F = -5a + 4a \)
    The factors (F) should give a product (P) and sum (S).
    Replace -a with -5a + 4a so that the terms become 4.
    \( = a^2 - 5a + 4a - 20 \\ = a(a - 5) + 4(a - 5) \\ = (a + 4)(a - 5) \)

Using the difference of two squares method

When two square numbers are subtracting each other like \( a^2 - b^2 \), they can be factorised by splitting \( a^2 \) and \( -b^2 \) into their respective factors.

  • \( a^2 - b^2 \)
    \( a^2 = a \times a \)
    \( -b^2 = (-)b \times (+)b \)
    therefore:
    \( a^2 - b^2 = (a - b)(a + b) \)

Example 8
Questions

  • Factorise the following expressions.
    1. \( x^2 - 49 \)
    2. \( 2x^2 - 32 \)
    3. \( 1 - 9x^2 \)

Answers

  1. \( x^2 - 49 \)
    \( x^2 = x \times x \)
    \( -49 = (-)7 \times (+)7 \)
    therefore:
    \( x^2 - 49 = (x - 7)(x + 7) \)
  2. \( 2x^2 - 32 \). Pull out the common factor 2.
    \( 2(x^2 - 16) \)
    \( x^2 = x \times x \)
    \( -16 = (-)4 \times (+)4 \)
    therefore:
    \( 2x^2 - 32 = 2(x - 4)(x + 4) \)
  3. \( 1 - 9x^2 \)
    \( 1 = 1 \times 1 \)
    \( -9x^2 = (-)3x \times (+)3x \)
    therefore:
    \( 1 - 9x^2 = (1 - 3x)(1 + 3x) \)

You may attempt the following exercise.

Exercise 2.2: Factorisation
Questions

  • Factorise the following.
    1. \( 3m^2 - 12m \)
    2. \( -10ab + 5a \)
    3. \( x^2 + 6x + 8 \)
    4. \( 3x^2 + 8x + 5 \)
    5. \( 3m^2 – 48 \)
    6. \( a^2 + 15a + 50 \)
    7. \( 3r^2 – 5r − 8 \)
    8. \( m^2 + 5mx − 3mx– 15x^2 \)
    9. \( 81m − 3m^3 \)
    10. \( a^2 - \dfrac{4}{9} \)

Answers

  1. 3m(m - 4)
  2. –5a(2b – 1)
  3. \( (x + 4)(x + 2) \)
  4. \( (x + 1)(3x + 5) \)
  5. \( 3(m - 4)(m + 4) \)
  6. \( (a + 5)(a + 10) \)
  7. \( (r + 1)(3r - 8) \)
  8. \( (m - 3x)(m + 5x) \)
  9. \( 3m(27 − m^2) \)
  10. \( (a - \dfrac{2}{3})(a + \dfrac{2}{3}) \)

2.3.3. Substitution

Substitution means replacing the variables (letters) in an algebraic expression with their numerical values. The value of the term or expression can then be worked out.

Example 9
Questions

  • Given that a = 3, b = 2 and c = −1 find the value of
    1. a + 3b
    2. abc
    3. \( \dfrac{a + b}{c} \)
    4. \( \dfrac{a - c}{b + c} \)
    5. \( (a - c)^b \)

Answers

  1. a + 3b
    = 3 + 3 × 2
    = 3 + 6
    = 9
  2. abc
    = 3 × 2 × −1
    = −6
  3. \( \dfrac{a + b}{c} \\ = \dfrac{3 + 2}{-1} \\ = \dfrac{5}{-1} \\ = -5 \)
  4. \( \dfrac{a - c}{b + c} \\ = \dfrac{3 - (-1)}{2 + (-1)} \\ = \dfrac{4}{1} \\ = 4 \)
  5. \( (a - c)^b \\ = (3 - (-1))^2 \\ = (3 + 1)^2 \\ = 4^2 = 16 \)

You may attempt the following exercise.

Exercise 2.3: Substitution
Questions

  1. Given that a = 4 and b = –1. Find the value of
    1. a + 3b
    2. \( a^2 - b^2 \)
    3. \( (a - b)^2 \)
  2. Given the expression \( 3x^2 - 4x + 5 \). Simplify the expression when
    1. x = 0
    2. x = 1
    3. x = –1
    4. x = 2
    5. x = –2
  3. If \( m = nq - \dfrac{q}{n} \)
    1. Find the value of m when n = 14 and \( q = \dfrac{2}{7} \)
    2. Find q when m = 8 and n = 3

Answers

    1. 1
    2. 15
    3. 25
    1. 5
    2. 4
    3. 12
    4. 9
    5. 25
    1. \( 3\dfrac{48}{49} \)
    2. 3

2.4. Algebraic fractions

Algebraic fractions can be written in their lowest terms, added, subtracted, multiplied and divided.

2.4.1. Lowest terms

To reduce an algebraic fraction to its lowest terms, factorise the numerator and the denominator then cancel out common factors.

Example 10
Questions

  • Simplify the following.
    1. \( \dfrac{a^2 + ab}{b^2 + ab} \)
    2. \( \dfrac{x^2 - 5x + 6}{x^2 - 3x + 2} \)
    3. \( \dfrac{a^2 - b^2}{a^2 - 2ab + b^2} \)

Answers

  1. \( \dfrac{a^2 + ab}{b^2 + ab} \\ = \dfrac{a(a + b)}{b(b + a)} \\ = \dfrac{a(a + b)}{b(a + b)} \\ = \dfrac{a}{b} \)
  2. \( \dfrac{x^2 - 5x + 6}{x^2 - 3x + 2} \\ = \dfrac{(x - 3)(x - 2)}{(x - 2)(x - 1)} \\ = \dfrac{x - 3}{x - 1} \)
  3. \( \dfrac{a^2 - b^2}{a^2 - 2ab + b^2} \\ = \dfrac{(a - b)(a + b)}{(a - b)(a - b)} \\ = \dfrac{a + b}{a - b} \)

Note: To be able to reduce algebraic fractions to lowest terms you should know factorisation well.

2.4.2. Addition and subtraction of algebraic fractions

Algebraic fractions can be added or subtracted by putting the fractions under a common denominator, then simplifying.

Example 11
Questions

  • Simplify the following.
    1. \( \dfrac{3}{2x + 2y} + \dfrac{4}{3x + 3y} \)
    2. \( \dfrac{m + 4}{2m - 8} - \dfrac{m + 3}{12 - 3m} \)

Answers

  1. \( \dfrac{3}{2x + 2y} + \dfrac{4}{3x + 3y} \)
    factorise the denominator
    \( = \dfrac{3}{2(x + y)} + \dfrac{4}{3(x + y)} \)
    find LCM of 2(x + y) and 3(x + y), which is 6(x + y)
    then put the fractions under one denominator, and simplify
    \( = \dfrac{3 \times 3 + 2 \times4}{6(x + y)} = \dfrac{17}{6(x + y)} \)
  2. \( \dfrac{m + 4}{2m - 8} - \dfrac{m + 3}{12 - 3m} \)
    Find the LCM of 2m − 8 and 12 − 3m by first factorising the 2 expressions
    2m − 8 = 2(m − 4)
    12 − 3m = −3(m − 4)
    therefore LCM = −6(m − 4)
    then put the fractions under one denominator, and simplify
    \( = \dfrac{-3(m + 4) - 2(m + 3)}{-6(m - 4)} \\ = \dfrac{-3m - 12 - 2m - 6}{-6(m - 4)} \\ = \dfrac{5m + 18}{6(m - 4)} \)

You may attempt the following exercise.

Exercise 2.4: Addition and subtraction of algebraic fractions
Questions

  • Simplify the following.
    1. \( 6 - \dfrac{m + n}{c} \)
    2. \( \dfrac{7}{x - 2} - \dfrac{3}{x - 3} \)
    3. \( \dfrac{5}{2(a + b)} - \dfrac{4}{3(a + b)} \)
    4. \( \dfrac{3}{a^2 - 2a -8} + \dfrac{5}{a^2 - 6a + 8} \)
    5. \( \dfrac{7}{x^2 - 16} + \dfrac{3}{(x + 4)^2} \)

Answers

  1. \( \dfrac{6c - m - n}{c} \)
  2. \( \dfrac{4x - 15}{(x - 2)(x - 3)} \)
  3. \( \dfrac{7}{6(a + b)} \)
  4. \( \dfrac{8a + 4}{(a - 4)(a - 2)(a + 2)} \)
  5. \( \dfrac{10x + 16}{(x - 4)(x + 4)^2} \)

2.4.3. Multiplication and division of algebraic fractions

To multiply or divide algebraic expressions, you should be able to reduce the fractions using common factors. Remember multiplication and division of fractions:

  • \( \dfrac{a}{b} \times \dfrac{m}{n} \\ = \dfrac{am}{bn} \)
  • \( \dfrac{a}{b} \div \dfrac{m}{n} \\ = \dfrac{a}{b} \times \dfrac{n}{m} \\ = \dfrac{an}{bm} \)

Example 12
Questions

  • Simply the following.
    1. \( \dfrac{a^2 - 9}{a^2 + 5a + 6} \times \dfrac{a^2 + 2a}{a^2} \)
    2. \( \dfrac{a^2 - b^2}{a^2 - 2ab + b^2} \div \dfrac{a^2 + ab}{b^2 - ab} \)

Answers

  1. \( \dfrac{a^2 - 9}{a^2 + 5a + 6} \times \dfrac{a^2 + 2a}{a^2} \\ = \dfrac{(a - 3)(a + 3)}{(a + 3)(a + 2)} \times \dfrac{a(a + 2)}{a^2} \\ = \dfrac{a(a - 3)}{a^2} \\ = \dfrac{a - 3}{a} \)
  2. \( \dfrac{a^2 - b^2}{a^2 - 2ab + b^2} \div \dfrac{a^2 + ab}{b^2 - ab} \\ = \dfrac{(a - b)(a + b)}{(a - b)(a - b)} \div \dfrac{a(a + b)}{b(b - a)} \\ = \dfrac{(a - b)(a + b)}{(a - b)(a - b)} \times \dfrac{b(b - a)}{a(a + b)} \\ = \dfrac{b(b - a)}{a(a - b)} \\ = \dfrac{-b(a - b)}{a(a - b)} \\ = \dfrac{-b}{a} \)

You may attempt the following exercise.

Exercise 2.5: Multiplication and division of algebraic fractions
Questions

  • Simplify the following.
    1. \( \dfrac{m^2 + mn}{m^2 - n^2} \)
    2. \( \dfrac{15 + 2a - a^2}{a^2 - 25} \)
    3. \( \dfrac{d^2 - 3d - 4}{d^2 - 4d} \div \dfrac{d^2 - 4d + 4}{d^2 - 4} \)
    4. \( \dfrac{x^2 - y^2}{xy + x^2} \times \dfrac{2x^2}{xy - x^2} \)
    5. \( \dfrac{3}{a + 5} \times \dfrac{a^2 - 25}{9} \)

Answers

  1. \( \dfrac{m}{m - n} \)
  2. \( \dfrac{-a - 3}{a + 5} \)
  3. \( \dfrac{(d + 1)(d + 2)}{d(d - 2)} \)
  4. \( -2 \)
  5. \( \dfrac{a - 5}{3} \)

2.5. Summary

From this chapter you have learnt how to simplify different types of algebraic expressions, that is factorisation, finding LCM and HCF and simplification of algebraic expressions. The skills you have mastered will help you understand what is in store for you in the next chapter.

2.6. Further reading

  1. Macrae, M. F., Madungwe, L. and Mutangadura, A. (2017). New General Mathematics Book 3. Pearson Capetown.
  2. Macrae, M. F., Madungwe, L. and Mutangadura, A. (2017). New General Mathematics Book 4. Pearson Capetown.
  3. Meyers, C., Graham, B., Dawe, L. (2004). Mathscape Working Mathematically. 9th Edition. MacMillan.
  4. Pimentel, R. and Wall, T. (2011). International Mathematics. Hodder UK.
  5. Rayner, D. (2005). Extended Mathematics. Oxford New York.

2.7. Test 2

Answer the following questions. Do not use a calculator.

Questions

  1. Simplify the following expressions.
    1. 5x - (3x - 2)   [2]
    2. –3(2x - 3y + 5)   [2]
    3. \( (2x + 3)^2 \)   [2]
  2. Write the following as single fractions.
    1. \( \dfrac{2x - 1}{3} + \dfrac{5x + 2}{4} \)   [2]
    2. \( \dfrac{3x + 4}{3} - \dfrac{7x + 1}{5} \)   [2]
    3. \( \dfrac{5}{x + 2} - \dfrac{3}{x - 3} \)   [2]
  3. Express the following fractions in their lowest terms.
    1. \( \dfrac{2x - 2y}{2y - 2x} \)   [2]
    2. \( \dfrac{a^2 + 5a + 6}{a^2 - 9} \)   [2]
    3. \( \dfrac{x^2 - 4}{x^2 - 6x + 8} \)   [2]
  4. Factorise completely.
    1. \( 125x^3 - 5x \)   [2]
    2. \( 2x^2 - 25x - 27 \)   [2]
    3. \( a^2 + am - an - mn \)   [2]
  5. Simplyfy the following.
    1. \( \dfrac{m + 4}{m^2 + 8m + 15} \times \dfrac{m^2 + 2m - 3}{m^2 - 16} \)   [3]
    2. \( \dfrac{a^2 - b^2}{a^2 - 2ab + b^2} \div \dfrac{a^2 + ab}{b^2 - ab} \)   [3]
  6. The sides of a rectangle are (3x - 4)cm and (2x + 1)cm.
    1. Calculate the perimeter of the rectangle in terms of x in its simplest form.   [2]
    2. Calculate the area of the rectangle in terms of x in its simplest form.   [2]
    3. If (3x - 4)cm and (2x + 1)cm are sides of a square pipe, find the value of x.   [2]

Answers

    1. 2x + 2
    2. –6x + 9y − 15
    3. \( 4x^2 + 12x + 9 \)
    1. \( \dfrac{23x + 2}{12} \)
    2. \( \dfrac{17 - 6x}{15} \)
    3. \( \dfrac{2x - 21}{(x + 2)(x - 3)} \)
    1. \( -1 \)
    2. \( \dfrac{a + 2}{a - 3} \)
    3. \( \dfrac{x + 2}{x - 4} \)
    1. 5x(5x - 1)(5x + 1)
    2. (x + 1)(2x - 27)
    3. (a - n)(a + m)
    1. \( \dfrac{m - 1}{(m - 4)(m + 5)} \)
    2. \( \dfrac{-b}{a} \)
    1. \( 16x - 6 \)
    2. \( 6x^2 - 5x - 4 \)
    3. \( 5 \)