Section B — Chapters 23–39

Algebra: Further Expansions and Division

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to expand algebraic expressions by multiplying a term by a bracket.
✓By the end of this lesson students will be able to simplify expressions involving multiplication of a term by a bracket.
✓By the end of this lesson students will be able to divide algebraic terms.
✓By the end of this lesson students will be able to divide algebraic expressions by a single term.

Key concepts

Multiplying a term by a bracket

When a term is multiplied by an expression inside a bracket, the term outside the bracket must be multiplied by *every* term inside the bracket. This is known as the distributive law. For example, a(b + c) means 'a multiplied by b' PLUS 'a multiplied by c'.

a(b + c) = ab + ac

Division in Algebra

To divide algebraic terms, we divide the numerical coefficients first, and then divide the variables. When dividing variables with powers, we subtract the powers. For example, x^m divided by x^n is x raised to the power of (m - n).

x^m / x^n = x^(m-n) (where m is greater than or equal to n)

Key facts to remember

1The distributive law states that a(b + c) = ab + ac.
2When multiplying terms, multiply the numerical coefficients and then multiply the variables.
3When multiplying variables with powers, add the powers (e.g., x^m * x^n = x^(m+n)).
4When dividing variables with powers, subtract the powers (e.g., x^m / x^n = x^(m-n)).
5A negative number multiplied or divided by a negative number gives a positive result.
6A negative number multiplied or divided by a positive number gives a negative result.

Worked examples

Example 1

Expand 5(x + 3).

IMultiply 5 by x: 5 * x = 5x

IIMultiply 5 by 3: 5 * 3 = 15

IIICombine the terms: 5x + 15

Answer

5x + 15

Example 2

Expand -4(2y - 6).

IMultiply -4 by 2y: -4 * 2y = -8y

IIMultiply -4 by -6: -4 * -6 = +24

IIICombine the terms: -8y + 24

Answer

-8y + 24

Be very careful with negative signs. A negative multiplied by a negative gives a positive.

Example 3

Simplify 18a^4 / 6a^2.

IDivide the numerical coefficients: 18 / 6 = 3

IIDivide the variables: a^4 / a^2 = a^(4-2) = a^2

IIICombine the results: 3a^2

Answer

3a^2

Example 4

Simplify (20x^3 - 10x^2) / 5x.

IDivide the first term by 5x: 20x^3 / 5x = (20/5) * (x^3/x) = 4x^2

IIDivide the second term by 5x: -10x^2 / 5x = (-10/5) * (x^2/x) = -2x

IIICombine the results: 4x^2 - 2x

Answer

4x^2 - 2x

Remember to divide *each* term in the numerator by the term in the denominator.

Common mistakes

✗Only multiplying the first term inside the bracket by the term outside, forgetting to multiply all other terms.
✗Making sign errors when multiplying or dividing negative numbers.
✗Incorrectly applying the rules for powers, especially when subtracting powers during division.
✗When dividing an expression by a single term, only dividing the first term and forgetting to divide all other terms in the numerator.

Exam tips

★Always show your working step-by-step to avoid errors and gain partial marks, especially with signs.
★Pay close attention to positive and negative signs throughout your calculations; a small error can lead to a completely wrong answer.
★If unsure, write out the multiplication or division for each term separately before combining them.
★Practise regularly to become confident with the rules for powers and signs, as these are fundamental to algebra.

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