Algebra

Algebra: Notation, Simplifying and Expanding

Year 10 · Year 11

✓Expand algebraic expressions involving single, double, and triple brackets.
✓Factorise algebraic expressions, including quadratics and expressions involving common factors.
✓Apply the laws of indices to simplify algebraic expressions with integer, fractional, and negative powers.
✓Simplify algebraic expressions by collecting like terms.

Key concepts

Expanding Brackets

Expanding brackets means multiplying out the terms inside the brackets by the term(s) outside. This process removes the brackets from the expression, converting a product into a sum or difference of terms.

Single bracket: a(b + c) = ab + ac Double brackets (FOIL method): (a + b)(c + d) = ac + ad + bc + bd Triple brackets: (a + b)(c + d)(e + f) - first expand two brackets, then multiply the result by the third bracket.

Factorising

Factorising is the reverse process of expanding. It involves writing an expression as a product of its factors, often by taking out a common factor or by finding two binomials that multiply to give the original expression. This is a key skill for solving equations and simplifying fractions.

Common factor: ab + ac = a(b + c) Quadratic (x^2 + bx + c): Find two numbers that multiply to 'c' and add to 'b'. Difference of two squares: a^2 - b^2 = (a - b)(a + b)

Laws of Indices (Index Laws)

Index laws are a set of rules used to simplify expressions involving powers (indices). They provide a systematic way to manipulate terms with exponents.

Multiplication: a^m \times a^n = a^{m+n} Division: a^m \div a^n = a^{m-n} Power of a power: (a^m)^n = a^{mn} Zero index: a^0 = 1 (for a \neq 0) Negative index: a^{-n} = \frac{1}{a^n} Fractional index: a^{\frac{1}{n}} = \sqrt[n]{a} Fractional index: a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}

Key facts to remember

1Expanding means multiplying out brackets to remove them. Factorising means putting expressions into brackets by finding common factors; they are inverse operations.
2The distributive law, a(b + c) = ab + ac, is the fundamental principle for expanding single brackets.
3For double brackets, use the FOIL method (First, Outer, Inner, Last) or a grid method to ensure all four products are calculated and then combined.
4Memorise the special case for factorising the difference of two squares: a^2 - b^2 = (a - b)(a + b).
5All index laws must be known: a^m \times a^n = a^{m+n}, a^m \div a^n = a^{m-n}, (a^m)^n = a^{mn}, a^0 = 1, a^{-n} = \frac{1}{a^n}, a^{\frac{m}{n}} = (\sqrt[n]{a})^m.
6When expanding triple brackets, always expand two brackets first to get a quadratic expression, then multiply this result by the remaining linear bracket.

Worked examples

Example 1

Expand and simplify (2x - 3)(x + 5).

IApply the distributive law (or FOIL method):

II= 2x(x + 5) - 3(x + 5)

IIIMultiply out each term:

IV= 2x^2 + 10x - 3x - 15

VCollect like terms (the 'x' terms):

VI= 2x^2 + 7x - 15

Answer

2x^2 + 7x - 15

The FOIL method (First, Outer, Inner, Last) is a common mnemonic for expanding double brackets: (2x)(x) + (2x)(5) + (-3)(x) + (-3)(5).

Example 2

Factorise 3x^2 + 11x - 4.

IIdentify two numbers that multiply to (3 \times -4 = -12) and add to 11. These numbers are 12 and -1.

IIRewrite the middle term (11x) using these two numbers:

III= 3x^2 + 12x - x - 4

IVFactorise by grouping the first two terms and the last two terms:

V= 3x(x + 4) - 1(x + 4)

VITake out the common bracket (x + 4):

VII= (3x - 1)(x + 4)

Answer

(3x - 1)(x + 4)

Always check your factorisation by expanding the brackets to ensure it matches the original expression.

Example 3

Simplify \frac{(y^3)^2 \times y^5}{y^4}.

IApply the power of a power rule (a^m)^n = a^{mn} to the numerator:

II(y^3)^2 = y^{3 \times 2} = y^6

IIIThe expression becomes: \frac{y^6 \times y^5}{y^4}

IVApply the multiplication rule a^m \times a^n = a^{m+n} to the numerator:

Vy^6 \times y^5 = y^{6+5} = y^{11}

VIThe expression becomes: \frac{y^{11}}{y^4}

VIIApply the division rule a^m \div a^n = a^{m-n}:

VIIIy^{11-4} = y^7

Answer

y^7

Work systematically, applying one index law at a time to avoid errors. Ensure the base is the same for multiplication and division rules.

Example 4

Expand and simplify (x + 1)(x - 2)(x + 3).

IFirst, expand the first two brackets, for example, (x + 1)(x - 2):

II= x(x - 2) + 1(x - 2)

III= x^2 - 2x + x - 2

IV= x^2 - x - 2

VNow, multiply this quadratic result by the third bracket (x + 3):

VI= (x^2 - x - 2)(x + 3)

VIIMultiply each term in the first bracket by each term in the second:

VIII= x^2(x + 3) - x(x + 3) - 2(x + 3)

9= x^3 + 3x^2 - x^2 - 3x - 2x - 6

10Collect all like terms:

11= x^3 + (3x^2 - x^2) + (-3x - 2x) - 6

12= x^3 + 2x^2 - 5x - 6

Answer

x^3 + 2x^2 - 5x - 6

It is crucial to be careful with signs and to collect all like terms accurately in the final step.

Common mistakes

✗Sign errors: Incorrectly applying negative signs when expanding or factorising, leading to incorrect terms.
✗Squaring a binomial incorrectly: Expanding (x + y)^2 as x^2 + y^2 instead of the correct x^2 + 2xy + y^2.
✗Incomplete factorisation: Not taking out the highest common factor, or not fully factorising a quadratic expression.
✗Misapplying index laws: Confusing addition/subtraction of powers with multiplication/division of bases (e.g., thinking a^m + a^n = a^{m+n}).
✗Errors with negative or fractional indices: Incorrectly handling expressions like a^{-n} or a^{1/n} (e.g., writing a^{-n} as -a^n).

Exam tips

★Show all working: For expanding and factorising, examiners award marks for clear, step-by-step methods, not just the final answer.
★Check your answers: After factorising, expand your answer to see if it matches the original expression. For expanding, substitute a simple value for 'x' into both the original and expanded expression to check for consistency.
★Be methodical: For complex expansions or index law problems, tackle one step at a time to avoid errors and ensure accuracy.
★Practise index laws regularly: They are often combined with other algebra topics, so quick recall and accurate application are crucial for efficiency and correctness.

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