Strand 4 — Algebra

Algebraic Expressions

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓By the end of this lesson students will be able to simplify algebraic expressions by collecting like terms.
✓By the end of this lesson students will be able to expand expressions by multiplying out brackets using the Distributive Law.
✓By the end of this lesson students will be able to factorise expressions using common factors.
✓By the end of this lesson students will be able to factorise expressions by grouping terms.
✓By the end of this lesson students will be able to factorise quadratic expressions of the form ax² + bx + c.
✓By the end of this lesson students will be able to factorise the difference of two squares.

Key concepts

Algebraic Expression

An algebraic expression is a combination of numbers, variables (letters), and mathematical operations (+, -, ×, ÷). It does not contain an equals sign.

Term

A term is a single number, a single variable, or a product of numbers and variables. Terms are separated by addition or subtraction signs.

Like Terms

Like terms are terms that have the exact same variables raised to the exact same powers. Only like terms can be added or subtracted.

Multiplying out brackets (Distributive Law)

To multiply out brackets, each term inside the bracket is multiplied by the term outside the bracket.

a(b + c) = ab + ac

Factorising

Factorising is the reverse process of multiplying out brackets. It involves writing an expression as a product of its factors.

Factorising by Common Factor

To factorise by a common factor, find the greatest common factor (GCF) of all terms in the expression and write it outside a bracket, with the remaining terms inside.

ab + ac = a(b + c)

Factorising by Grouping

This method is typically used for expressions with four terms. Group terms in pairs, find a common factor for each pair, and then find a common binomial factor.

ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

Factorising Quadratic Expressions (ax² + bx + c)

For a quadratic expression where a=1 (x² + bx + c), find two numbers that multiply to 'c' and add to 'b'. For a ≠ 1, a systematic approach (like the AC method or trial and error) is used.

x² + (p+q)x + pq = (x + p)(x + q)

Difference of Two Squares

An expression of the form a² - b² can always be factorised into the product of two binomials.

a² - b² = (a - b)(a + b)

Key facts to remember

1Like terms have the same variables raised to the same powers.
2The Distributive Law states a(b + c) = ab + ac.
3Factorising is the reverse process of multiplying out brackets.
4Always look for a common factor as the first step in any factorising problem.
5The difference of two squares formula is a² - b² = (a - b)(a + b).
6When factorising x² + bx + c, find two numbers that multiply to c and add to b.
7When factorising ax² + bx + c (a ≠ 1), use the AC method or trial and error.
8An expression is fully factorised when no more common factors can be taken out of any of its factors.

Worked examples

Example 1

Simplify the expression: 3(2x - 5) - 2(x + 4) + 7x

IMultiply out the first bracket: 3(2x) - 3(5) = 6x - 15

IIMultiply out the second bracket, being careful with the negative sign: -2(x) - 2(4) = -2x - 8

IIIRewrite the expression: 6x - 15 - 2x - 8 + 7x

IVGroup like terms together: (6x - 2x + 7x) + (-15 - 8)

VCombine the like terms: (6 - 2 + 7)x + (-23)

Answer

11x - 23

Remember to distribute the negative sign to ALL terms inside the second bracket.

Example 2

Factorise fully: (a) 12x²y - 18xy² (b) 3ab - 6a + 5b - 10

I(a) Find the greatest common factor (GCF) of 12x²y and 18xy².

IIThe GCF of the numbers (12, 18) is 6.

IIIThe GCF of the variables (x², x) is x.

IVThe GCF of the variables (y, y²) is y.

VSo, the overall GCF is 6xy.

VIDivide each term by the GCF: (12x²y) / (6xy) = 2x and (-18xy²) / (6xy) = -3y.

VII(b) Group the terms in pairs: (3ab - 6a) + (5b - 10)

VIIIFactorise the first pair: 3a(b - 2)

9Factorise the second pair: +5(b - 2)

10Notice the common binomial factor (b - 2).

11Factor out the common binomial.

Answer

(a) 6xy(2x - 3y) (b) (3a + 5)(b - 2)

Always check for a common factor first, even before grouping.

Example 3

Factorise fully: (a) x² + 7x + 12 (b) 49p² - 81q² (c) 3x² + 10x - 8

I(a) Find two numbers that multiply to 12 and add to 7. These numbers are 3 and 4.

II(b) Recognise this as the difference of two squares: (7p)² - (9q)².

IIIApply the formula a² - b² = (a - b)(a + b).

IV(c) For 3x² + 10x - 8, multiply a by c: 3 × (-8) = -24.

VFind two numbers that multiply to -24 and add to 10. These numbers are 12 and -2.

VIRewrite the middle term (10x) using these numbers: 3x² + 12x - 2x - 8

VIIFactorise by grouping the first two terms and the last two terms: 3x(x + 4) - 2(x + 4)

VIIIFactor out the common binomial (x + 4).

Answer

(a) (x + 3)(x + 4) (b) (7p - 9q)(7p + 9q) (c) (x + 4)(3x - 2)

For quadratic factorising, always check your answer by multiplying out the brackets.

Common mistakes

✗Incorrectly identifying like terms (e.g., treating x and x² as like terms).
✗Making sign errors when multiplying out brackets, especially with negative signs outside the bracket.
✗Forgetting to multiply *all* terms inside the bracket by the term outside.
✗Not factorising fully (e.g., leaving a common factor inside a bracket).
✗Confusing the difference of two squares (a² - b²) with the sum of two squares (a² + b²), which cannot be factorised over real numbers.

Exam tips

★Show all your steps clearly, especially when multiplying out brackets or factorising. This helps you earn partial marks even if your final answer is incorrect.
★Always check your factorised answers by multiplying out the brackets. If you get the original expression, your factorisation is correct.
★Be meticulous with positive and negative signs. A single sign error can lead to a completely wrong answer.
★Practice all types of factorisation regularly to build speed and accuracy.

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