Cambridge Lower Secondary CheckpointStage 6

Algebra (6Ae)

Mathematics Stage 6 Chapter Notes

What this chapter covers

Algebra — Expressions, equations and formulae
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Algebra (6Ae) notes

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In this chapter

  1. 1Expanding Single and Double Brackets
  2. 2Factorising: Finding Common Factors
  3. 3Factorising Quadratics: x² + bx + c
  4. 4Factorising the Difference of Two Squares
  5. 5Factorising Harder Quadratics: ax² + bx + c
  6. 6Simplifying Algebraic Fractions
  7. 7Adding and Subtracting Algebraic Fractions

1. Expanding Single and Double Brackets

Expanding brackets means to multiply out the terms. For a single bracket, you multiply the term outside by everything inside. This is called the distributive law. For example, a(b +c) becomes ab + ac. For double brackets, like (a + b)(c + d), a useful method is FOIL: multiply the First terms (ac), Outer terms (ad), Inner terms (bc), and Last terms (bd). Then, collect any like terms to simplify. A special case is squaring a binomial, like (x + y)², which expands to x² + 2xy + y², not just x² + y².

a(b + c) = ab + ac

(a + b)(c + d) = ac + ad + bc + bd

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Key term

Expand: To multiply out the terms within brackets to remove them from an expression.

Common pitfall

When squaring a binomial like (x - 5)², many students incorrectly write x² - 25 or x² + 25, completely forgetting the middle term, which comes from 2 * x * (-5) = -10x. The correct expansion is x² - 10x + 25.

Worked example 12 marks

Expand and simplify 5x(3x - 7).

  1. 1

    Multiply the term outside the bracket, 5x, by the first term inside, 3x.

  2. 2

    5x * 3x = 15x²

  3. 3

    Multiply the term outside, 5x, by the second term inside, -7.

  4. 4

    5x * (-7) = -35x

  5. 5

    Combine the results.

  6. 6

    Answer: 15x² - 35x

Worked example 22 marks

Expand and simplify (y + 5)(y - 8).

  1. 1

    Use the FOIL method.

  2. 2

    First: y * y = y²

  3. 3

    Outer: y * (-8) = -8y

  4. 4

    Inner: 5 * y = 5y

  5. 5

    Last: 5 * (-8) = -40

  6. 6

    Combine the terms: y² - 8y + 5y - 40

  7. 7

    Collect the like terms (-8y + 5y = -3y).

  8. 8

    Answer: y² - 3y - 40

Worked example 33 marks

Expand and simplify (2x - 3)³.

  1. 1

    First, expand (2x - 3)². This is (2x - 3)(2x - 3).

  2. 2

    (2x)² + 2(2x)(-3) + (-3)² = 4x² - 12x + 9

  3. 3

    Now, multiply this result by (2x - 3): (4x² - 12x + 9)(2x - 3)

  4. 4

    Multiply every term in the first bracket by 2x: 2x(4x² - 12x + 9) = 8x³ - 24x² + 18x

  5. 5

    Multiply every term in the first bracket by -3: -3(4x² - 12x + 9) = -12x² + 36x - 27

  6. 6

    Combine the results: 8x³ - 24x² + 18x - 12x² + 36x - 27

  7. 7

    Collect like terms: 8x³ + (-24x² - 12x²) + (18x + 36x) - 27

  8. 8

    Answer: 8x³ - 36x² + 54x - 27

Recap

  • To expand a single bracket, multiply the outside term by every term inside.
  • Use the FOIL method for expanding double brackets: First, Outer, Inner, Last.
  • Always collect like terms after expanding to simplify the expression.
  • Remember the pattern for squared brackets: (a ± b)² = a² ± 2ab + b².

Quick check

  1. Expand 3(2x - 9).1 mark
  2. Expand (x + 2)(x + 5).2 marks

2. Factorising: Finding Common Factors

Factorising is the reverse of expanding brackets. The simplest way to factorise is to find the highest common factor (HCF) of all the terms in an expression. The HCF is the largest number and highest power of each variable that divides into all the terms. You then 'pull out' this HCF and write what's left inside a bracket.

ab + ac = a(b + c)

Key term

Factorise: To write an algebraic expression as a product of its factors.

Examiner insight

Examiners look for complete factorisation. After taking out a common factor, always check if the expression remaining in the brackets can be factorised further, for example, into a difference of two squares.

Worked example 12 marks

Factorise completely 8x² + 12x.

  1. 1

    Find the HCF of the numbers 8 and 12. The HCF is 4.

  2. 2

    Find the HCF of the variables x² and x. The HCF is x.

  3. 3

    So, the overall HCF of the expression is 4x.

  4. 4

    Write the HCF outside a bracket: 4x(...).

  5. 5

    Divide the first term by the HCF: 8x² / 4x = 2x.

  6. 6

    Divide the second term by the HCF: 12x / 4x = 3.

  7. 7

    Place these results inside the bracket.

  8. 8

    Answer: 4x(2x + 3)

Worked example 22 marks

Factorise completely 15a³b² - 20a²b⁴.

  1. 1

    HCF of numbers 15 and 20 is 5.

  2. 2

    HCF of variables a³ and a² is a² (the lowest power).

  3. 3

    HCF of variables b² and b⁴ is b² (the lowest power).

  4. 4

    The overall HCF is 5a²b².

  5. 5

    Write the HCF outside a bracket: 5a²b²(...).

  6. 6

    Divide the first term: 15a³b² / 5a²b² = 3a.

  7. 7

    Divide the second term: -20a²b⁴ / 5a²b² = -4b².

  8. 8

    Place these inside the bracket.

  9. 9

    Answer: 5a²b²(3a - 4b²)

Recap

  • Factorising means putting an expression back into brackets.
  • Always look for the highest common factor (HCF) first.
  • The HCF includes the largest number and the highest power of variables that go into every term.
  • To check your answer, expand the brackets; you should get the original expression.

Quick check

  1. Factorise 10y² - 5y.2 marks

3. Factorising Quadratics: x² + bx + c

A quadratic trinomial of the form x² + bx + c can be factorised into two brackets: (x + p)(x + q). Your goal is to find two numbers, p and q, that satisfy two conditions: they must multiply to give the constant term 'c' (p × q =c) and add to give the coefficient of x, which is 'b' (p + q = b). It's helpful to list the factor pairs of 'c' and then see which pair adds up to 'b'. Pay close attention to the signs.

x² + (p + q)x + pq = (x + p)(x + q)

Key term

Quadratic Trinomial: An expression with three terms, of the form ax² + bx + c, where the highest power of the variable is 2.

Common pitfall

Students often find the correct pair of numbers but make sign errors in the brackets. For x² - 2x - 15, the numbers are -5 and +3. It's crucial to write (x - 5)(x + 3) and not, for example, (x + 5)(x - 3).

Worked example 12 marks

Factorise x² + 8x + 15.

  1. 1

    We need two numbers that multiply to make +15 and add to make +8.

  2. 2

    List factor pairs of 15: (1, 15), (3, 5), (-1, -15), (-3, -5).

  3. 3

    Check which pair adds to +8: 3 + 5 = 8.

  4. 4

    The numbers are +3 and +5.

  5. 5

    Write the expression as two brackets.

  6. 6

    Answer: (x + 3)(x + 5)

Worked example 22 marks

Factorise y² - 3y - 10.

  1. 1

    We need two numbers that multiply to make -10 and add to make -3.

  2. 2

    List factor pairs of -10: (1, -10), (-1, 10), (2, -5), (-2, 5).

  3. 3

    Check which pair adds to -3: 2 + (-5) = -3.

  4. 4

    The numbers are +2 and -5.

  5. 5

    Write the expression as two brackets.

  6. 6

    Answer: (y + 2)(y - 5)

Recap

  • To factorise x² + bx + c, find two numbers that multiply to 'c' and add to 'b'.
  • If 'c' is positive, the two numbers have the same sign (both positive if 'b' is positive, both negative if 'b' is negative).
  • If 'c' is negative, the two numbers have different signs.
  • Place these two numbers into brackets like (x + p)(x + q).

Quick check

  1. Factorise x² - 9x + 20.2 marks

4. Factorising the Difference of Two Squares

This is a special and very useful factorisation pattern. It applies to expressions with two terms, where one term is subtracted from the other, and both terms are perfect squares. An expression like a² - b² can be immediately factorised into (a - b)(a + b). To use this, identify what is being squared in each term (the 'a' and the 'b') and then put them into the two brackets, one with a minus and one with a plus.

a² - b² = (a - b)(a + b)

Key term

Difference of Two Squares: An expression with two squared terms where one is subtracted from the other.

Fun fact

This method is a mental maths trick for multiplying numbers close to a round number. For example, 29 × 31 = (30 - 1)(30 + 1) = 30² - 1² = 900 - 1 = 899.

Worked example 12 marks

Factorise x² - 81.

  1. 1

    Check if it's a difference of two squares. Yes: x² is a square, 81 is 9², and there is a subtraction.

  2. 2

    Here, a = x and b = 9.

  3. 3

    Apply the formula (a - b)(a + b).

  4. 4

    Answer: (x - 9)(x + 9)

Worked example 22 marks

Factorise 25y² - 4.

  1. 1

    Check the pattern. 25y² is (5y)² and 4 is 2². There is a subtraction.

  2. 2

    Here, a = 5y and b = 2.

  3. 3

    Apply the formula (a - b)(a + b).

  4. 4

    Answer: (5y - 2)(5y + 2)

Worked example 33 marks

Factorise completely 3x³ - 75x.

  1. 1

    First, always look for a common factor. The HCF of 3x³ and 75x is 3x.

  2. 2

    Factor out the HCF: 3x(x² - 25).

  3. 3

    Now look at the expression in the bracket: x² - 25.

  4. 4

    This is a difference of two squares, where a = x and b = 5.

  5. 5

    Factorise the bracket: (x - 5)(x + 5).

  6. 6

    Combine everything for the final answer.

  7. 7

    Answer: 3x(x - 5)(x + 5)

Recap

  • Look for two perfect square terms separated by a minus sign.
  • The factors are the square roots of the terms, placed in two brackets.
  • One bracket has a plus sign, the other has a minus sign.
  • Always check for a common factor first before using this method.

Quick check

  1. Factorise 49p² - 1.2 marks

5. Factorising Harder Quadratics: ax² + bx + c

When the coefficient of x² (the 'a' value) is not 1, factorising is a bit more complex. A reliable method is to 'split the middle term'. First, find two numbers that multiply to give the product 'ac' (a ×c) and add to give 'b'. Then, rewrite the middle term 'bx' using these two numbers. This gives you an expression with four terms, which you can then factorise by grouping in pairs.

Key term

Factorising by Grouping: A technique used for four-term expressions where terms are paired up, and a common factor is taken from each pair to reveal a common binomial factor.

Examiner insight

This is a multi-step process. Show your working clearly: state the product 'ac', the two numbers you find, the split expression, and the grouping. Marks are often awarded for correct intermediate steps even if the final answer has an error.

Worked example 13 marks

Factorise 3x² + 10x + 8.

  1. 1

    This is in the form ax² + bx + c, with a=3, b=10, c=8.

  2. 2

    Calculate ac: 3 × 8 = 24.

  3. 3

    Find two numbers that multiply to 24 and add to 10. The numbers are 4 and 6.

  4. 4

    Split the middle term (10x) into 4x + 6x: 3x² + 4x + 6x + 8.

  5. 5

    Factorise the first pair by grouping: x(3x + 4).

  6. 6

    Factorise the second pair by grouping: +2(3x + 4).

  7. 7

    The two brackets must be the same. Now factor out the common bracket (3x + 4).

  8. 8

    Answer: (3x + 4)(x + 2)

Worked example 23 marks

Factorise 6y² - 5y - 4.

  1. 1

    Here a=6, b=-5, c=-4.

  2. 2

    Calculate ac: 6 × (-4) = -24.

  3. 3

    Find two numbers that multiply to -24 and add to -5. The numbers are +3 and -8.

  4. 4

    Split the middle term (-5y) into +3y - 8y: 6y² + 3y - 8y - 4.

  5. 5

    Factorise the first pair: 3y(2y + 1).

  6. 6

    Factorise the second pair. Be careful with the negative: -4(2y + 1).

  7. 7

    The brackets (2y + 1) are common.

  8. 8

    Factor out the common bracket.

  9. 9

    Answer: (2y + 1)(3y - 4)

Recap

  • For ax² + bx + c, first calculate the product 'ac'.
  • Find two numbers that multiply to 'ac' and add to 'b'.
  • Rewrite the 'bx' term using these two numbers.
  • Factorise the resulting four-term expression by grouping in pairs.

Quick check

  1. Factorise 2x² + 11x + 5.3 marks

6. Simplifying Algebraic Fractions

Simplifying algebraic fractions works just like simplifying numerical fractions: you cancel out common factors from the numerator and the denominator. The crucial first step is to factorise both the numerator and the denominator completely. Once everything is written as a product of factors, you can cancel any factor that appears on both the top and the bottom.

(ka)/(kb) = a/b, where k is a common factor.

Key term

Factor: A number or algebraic expression that divides another number or expression exactly.

Common pitfall

Students are often tempted to 'cancel' individual terms across a plus or minus sign. For example, in (x² + 4)/(x), you cannot cancel the 'x's. Cancellation is only possible for common factors that multiply the entire numerator and denominator.

Worked example 12 marks

Simplify (12x³y²) / (18x²y⁵).

  1. 1

    Simplify the numbers: 12/18 simplifies to 2/3.

  2. 2

    Simplify the x terms: x³/x² = x³⁻² = x.

  3. 3

    Simplify the y terms: y²/y⁵ = y²⁻⁵ = y⁻³ = 1/y³.

  4. 4

    Combine the parts: (2 *x) / (3 * y³).

  5. 5

    Answer: 2x / 3y³

Worked example 23 marks

Simplify fully (x² + 2x - 15) / (x² - 9).

  1. 1

    Factorise the numerator: x² + 2x - 15. We need two numbers that multiply to -15 and add to 2. These are +5 and -3. So, the numerator is (x + 5)(x - 3).

  2. 2

    Factorise the denominator: x² - 9. This is a difference of two squares. So, the denominator is (x - 3)(x + 3).

  3. 3

    Rewrite the fraction with the factored expressions: [(x + 5)(x - 3)] / [(x - 3)(x + 3)].

  4. 4

    Cancel the common factor (x - 3) from the top and bottom.

  5. 5

    Answer: (x + 5) / (x + 3)

Recap

  • To simplify an algebraic fraction, you must factorise first.
  • Factorise the numerator completely.
  • Factorise the denominator completely.
  • Cancel out any common factors that appear in both the numerator and the denominator.

Quick check

  1. Simplify (5x + 10) / (x² - 4).3 marks

7. Adding and Subtracting Algebraic Fractions

To add or subtract algebraic fractions, you must have a common denominator. First, find the lowest common denominator (LCD), which is the lowest common multiple of the individual denominators. Often, you need to factorise the denominators to find the LCD. Then, rewrite each fraction as an equivalent fraction with the LCD. Once they have the same denominator, you can add or subtract the numerators. Finally, simplify the resulting fraction if possible.

a/b + c/d = (ad + bc) / bd

Key term

Lowest Common Denominator (LCD): The smallest expression that is a multiple of all the denominators in a set of fractions.

Common pitfall

When subtracting an algebraic fraction, a frequent error is forgetting to apply the negative sign to every term in the numerator of the fraction being subtracted. For example, in 3/(x) - (x-2)/(y), the numerator becomes 3y - (x-2) = 3y - x + 2, not 3y - x - 2.

Worked example 13 marks

Express as a single fraction: 4/(x+1) + 3/(x-2).

  1. 1

    The LCD is the product of the two denominators: (x+1)(x-2).

  2. 2

    Rewrite the first fraction: 4/(x+1) = 4(x-2) / (x+1)(x-2).

  3. 3

    Rewrite the second fraction: 3/(x-2) = 3(x+1) / (x+1)(x-2).

  4. 4

    Combine over the common denominator: [4(x-2) + 3(x+1)] / (x+1)(x-2).

  5. 5

    Expand the numerator: [4x - 8 + 3x + 3] / (x+1)(x-2).

  6. 6

    Simplify the numerator: (7x - 5) / (x+1)(x-2).

  7. 7

    Answer: (7x - 5) / (x+1)(x-2)

Worked example 23 marks

Express as a single fraction: 5/(x-3) - 2/(x+4).

  1. 1

    The LCD is (x-3)(x+4).

  2. 2

    Combine over the LCD: [5(x+4) - 2(x-3)] / (x-3)(x+4).

  3. 3

    Expand the numerator, being careful with the negative sign: [5x + 20 - 2x + 6] / (x-3)(x+4).

  4. 4

    Simplify the numerator: (3x + 26) / (x-3)(x+4).

  5. 5

    Answer: (3x + 26) / (x-3)(x+4)

Worked example 34 marks

Express as a single fraction: 6/(x²-4) - 1/(x+2).

  1. 1

    First, factorise the denominators. x²-4 = (x-2)(x+2).

  2. 2

    The denominators are (x-2)(x+2) and (x+2). The LCD is (x-2)(x+2).

  3. 3

    The first fraction already has the LCD. The second needs to be multiplied by (x-2).

  4. 4

    Rewrite the expression: 6/((x-2)(x+2)) - 1(x-2)/((x-2)(x+2)).

  5. 5

    Combine the numerators: [6 - (x-2)] / ((x-2)(x+2)).

  6. 6

    Simplify the numerator: [6 - x + 2] / ((x-2)(x+2)) = (8 -x) / ((x-2)(x+2)).

  7. 7

    Answer: (8 -x) / (x² - 4)

Recap

  • To add or subtract algebraic fractions, find a lowest common denominator (LCD).
  • Factorise denominators first to help find the LCD.
  • Rewrite each fraction with the LCD.
  • When subtracting, be very careful to apply the negative sign to all terms in the second numerator.
  • Combine the numerators and simplify if possible.

Quick check

  1. Express 2/x + 5/(x+3) as a single fraction.3 marks

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Expand and simplify (3x - 2)(x + 4) - (x - 1)².4 marks
  2. Factorise completely 50x² - 2.3 marks
  3. Factorise 6x² - 11x - 10.3 marks
  4. Express as a single fraction in its simplest form: (2x² + 7x - 4) / (x² - 16).4 marks
  5. Express as a single fraction: 3/(x-4) - 2/(x+5).3 marks
  6. Solve the equation: 5/(x+1) = 2/(x-2).3 marks
  7. A rectangle has length (2x + 5) cm and width (x - 1) cm. The area of the rectangle is 60 cm². Show that 2x² + 3x - 65 = 0 and find the dimensions of the rectangle.5 marks
  8. Factorise x² - 8x + 15.2 marks
  9. Express as a single fraction in its simplest form: 4/(x+2) + (x-3)/(x²+x-2).4 marks
  10. The area of a trapezium is given by the formula A = 0.5(a+b)h. Rearrange the formula to make 'a' the subject.2 marks

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