Algebra Revision Notes

Core method

This is Higher-tier content, so solutions should communicate multi-step reasoning precisely. Expanding brackets uses the distributive law: multiply every term in one bracket by every term in the other. With two binomials this creates four products. Combine like terms only after all products, including their signs, have been written. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

For example, (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15. For (2x − 3)(x + 4), the products are 2x², 8x, −3x and −12, giving 2x² + 5x − 12. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.

What is Factorising?

Factorising means rewriting an algebraic expression as a product of its factors. This is the opposite of expanding brackets. When you expand a(b + c), you get ab + ac. When you factorise ab + ac, you look for the common factor a and write it as a(b + c). The key is to identify what numbers or variables are common to all terms in the expression.

Steps to Factorise an Expression

1. Identify all terms in the expression. For example, in 8x + 12, the terms are 8x and 12. 2. Find the Highest Common Factor (HCF) of the numerical coefficients. For 8 and 12, the HCF is 4. 3. Find the HCF of the variable parts. In 8x and 12, x is only in one term, so there is no common variable factor. If the terms were 8x^2 + 12x, the common variable factor would be x. 4. Write the HCF outside a bracket. 5. Divide each original term by the HCF and place the results inside the bracket. For 8x + 12, divide 8x by 4 to get 2x, and divide 12 by 4 to get 3. So, 4(2x + 3) is the factorised form. 6. Always check by expanding your answer to ensure it matches the original expression.

Factorising with Variables

When terms include variables, you apply the same principle. For example, to factorise 10y - 15: the HCF of 10 and 15 is 5. There are no common variables to both terms. So, we take 5 outside the bracket: 5(10y/5 - 15/5) which simplifies to 5(2y - 3). Another example: factorise 4a^2 + 6ab. The HCF of 4 and 6 is 2. The HCF of a^2 and ab is a. So, the overall HCF is 2a. Taking 2a outside the bracket gives 2a(4a^2 / 2a + 6ab / 2a), which simplifies to 2a(2a + 3b).

What is Factorisation?

Factorisation is the process of reversing expansion. Instead of multiplying out brackets, you are putting terms back into brackets by identifying and taking out common factors. For example, if you have the expression 2x + 4, both terms (2x and 4) have a common factor of 2. So, we can 'take out' the 2, leaving (x + 2) inside the bracket. This gives us the factorised form: 2(x + 2). This skill is essential for simplifying algebraic expressions and solving equations later on.

Finding the Greatest Common Factor (GCF)

The first and most important step in factorising is to find the Greatest Common Factor (GCF) of all the terms in your expression. This GCF can be a number, a variable, or both. For numerical parts, find the largest number that divides into all coefficients. For variable parts, find the highest power of each variable that is common to all terms. For example, in 6x³ + 9x², the GCF of 6 and 9 is 3. The GCF of x³ and x² is x². So, the overall GCF is 3x². You then divide each original term by the GCF to find what goes inside the bracket.

Steps for Factorising Expressions

1. Identify all terms in the algebraic expression. For example, in 10y + 15, the terms are 10y and 15. In 4ab - 8a, the terms are 4ab and -8a.
2. Find the GCF for the numerical coefficients. For 10 and 15, the GCF is 5. For 4 and 8, the GCF is 4.
3. Find the GCF for the variables. For 10y and 15, there is no common variable factor. For 4ab and 8a, the common variable factor is 'a'.
4. Combine these to find the overall GCF of the expression. For 10y + 15, the GCF is 5. For 4ab - 8a, the GCF is 4a.
5. Write the GCF outside an empty set of brackets. E.g., 5( ) or 4a( ).
6. Divide each original term by the GCF and write the results inside the brackets. For 10y + 15:

10y ÷ 5 = 2y 15 ÷ 5 = 3 So, 5(2y + 3). For 4ab - 8a: 4ab ÷ 4a = b -8a ÷ 4a = -2 So, 4a(b - 2).

1. Always check your answer by expanding the brackets to ensure you get the original expression back.

Factorising Quadratics (ax² + bx + c)

When factorising a quadratic expression where the coefficient of x² is not 1 (i.e., 'a' in ax² + bx + c is not 1), follow these steps:

1. Multiply 'a' by 'c'.
2. Find two numbers that multiply to 'ac' and add to 'b'.
3. Rewrite the middle term (bx) using these two numbers.
4. Factorise by grouping the first two terms and the last two terms separately.
5. You should find a common bracket that can be factored out.

Example: Factorise 6x² + 13x + 5.

1. a × c = 6 × 5 = 30.
2. Numbers that multiply to 30 and add to 13 are 3 and 10.
3. Rewrite: 6x² + 3x + 10x + 5.
4. Group: 3x(2x + 1) + 5(2x + 1).
5. Final factorisation: (2x + 1)(3x + 5).

Difference of Two Squares (DOTS)

The difference of two squares is a special type of factorisation. It follows the pattern: a² - b² = (a - b)(a + b). To use this, you must identify two terms that are perfect squares and are being subtracted from each other. The 'a' and 'b' can be numbers, variables, or even expressions.

Examples:

• x² - 9 = x² - 3² = (x - 3)(x + 3).
• 4y² - 25 = (2y)² - 5² = (2y - 5)(2y + 5).
• 1 - 16p² = 1² - (4p)² = (1 - 4p)(1 + 4p).
• (x + 1)² - 4 = (x + 1)² - 2² = ((x + 1) - 2)((x + 1) + 2) = (x - 1)(x + 3).

DOTS is often used to simplify algebraic fractions or solve quadratic equations.

Factorising by Grouping

Factorisation by grouping is used when an expression has four terms or when restructuring a quadratic expression (as shown above). The general idea is to group terms that share common factors, factorise each group, and then look for a common bracket.

Example: Factorise y³ + 4y² - 2y - 8.

1. Group the terms: (y³ + 4y²) - (2y + 8).
2. Factorise each group: y²(y + 4) - 2(y + 4).
3. Notice the common bracket (y + 4).
4. Factor out the common bracket: (y + 4)(y² - 2).

This technique is fundamental for solving certain cubic equations or simplifying more complex algebraic expressions.