Number and algebra I (AQA GCSE Further Maths): Revision Notes

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Simplifying expressions

Simplifying algebraic expressions is a fundamental skill in GCSE algebra. When you're asked to simplify an expression, your goal is to write it in its most compact and efficient form. This process involves several key techniques that work together to make expressions cleaner and easier to work with.

What does simplifying mean?

Simplifying an algebraic expression means writing it in its most compact form possible. This involves using various techniques such as collecting like terms, removing brackets, factorising expressions, and finding common denominators when dealing with fractions. The aim is to reduce the expression to its simplest possible form while maintaining its mathematical value.

infoNote

Think of simplifying as tidying up your mathematical work - you're making it as neat and organised as possible without changing what it actually means.

Collecting like terms

Like terms are terms that contain exactly the same combination of variables raised to the same powers. For example, 3a3a and 5a5a are like terms, as are 2xy2xy and −7xy-7xy. When simplifying expressions, you can add or subtract like terms by combining their coefficients.

Let's look at how this works in practice:

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Worked Example: Collecting Like Terms

Starting with: 3a+4b−2c+a−3b−c3a + 4b - 2c + a - 3b - c

Step 1: Group like terms together

  • aa terms: 3a+a=4a3a + a = 4a
  • bb terms: 4b−3b=b4b - 3b = b
  • cc terms: −2c−c=−3c-2c - c = -3c

Step 2: Write the simplified expression Final answer: 4a+b−3c4a + b - 3c

Expanding and simplifying brackets

When you have brackets in an expression, you need to multiply everything inside the brackets by the term outside. This process is called expanding. After expanding, you can then collect like terms to simplify further.

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Worked Example: Expanding and Simplifying

Simplify: 2(3x−4y)−3(x+2y)2(3x - 4y) - 3(x + 2y)

Step 1: Expand the first bracket 2(3x−4y)=2×3x−2×4y=6x−8y2(3x - 4y) = 2 \times 3x - 2 \times 4y = 6x - 8y

Step 2: Expand the second bracket
−3(x+2y)=−3×x+(−3)×2y=−3x−6y-3(x + 2y) = -3 \times x + (-3) \times 2y = -3x - 6y

Step 3: Combine the expanded terms 6x−8y−3x−6y=3x−14y6x - 8y - 3x - 6y = 3x - 14y

chatImportant

Common Mistake Alert! When you have a negative sign in front of brackets, you must multiply ALL terms inside by the negative number. Many students forget to apply the negative to every term.

Multiplying algebraic terms

When multiplying terms that contain variables, you multiply the numerical coefficients together and add the powers of like variables. This follows the index law that says when multiplying powers of the same base, you add the indices.

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Worked Example: Multiplying Algebraic Terms

Multiply: 3x2yz×2xy33x^2yz \times 2xy^3

Step 1: Multiply the numbers 3×2=63 \times 2 = 6

Step 2: Multiply the xx terms x2×x1=x3x^2 \times x^1 = x^3

Step 3: Multiply the yy terms
y1×y3=y4y^1 \times y^3 = y^4

Step 4: The zz term appears only once zz remains as zz

Final answer: 6x3y4z6x^3y^4z

Simplifying algebraic fractions

Algebraic fractions can be simplified by dividing both the numerator and denominator by their highest common factor. This works the same way as simplifying numerical fractions.

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Worked Example: Simplifying Algebraic Fractions

Simplify: 12d3b2c28ab5c\frac{12d^3b^2c^2}{8ab^5c}

Step 1: Find the HCF of the numbers HCF of 12 and 8 = 4

Step 2: Find the HCF of the variables

  • For aa: lowest power = a1a^1
  • For bb: lowest power = b2b^2
  • For cc: lowest power = c1c^1
  • dd only appears in numerator

Step 3: HCF = 4ab2c4ab^2c

Step 4: Divide numerator and denominator by HCF 12d3b2c28ab5c=3d3c2b3\frac{12d^3b^2c^2}{8ab^5c} = \frac{3d^3c}{2b^3}

Factorising expressions

Factorising is the reverse process of expanding brackets. It involves taking out the highest common factor from all terms in an expression and writing the expression as a product of this factor and another expression in brackets.

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Worked Example: Factorising

Factorise: 3a2b+6ab23a^2b + 6ab^2

Step 1: Find HCF of coefficients HCF of 3 and 6 = 3

Step 2: Find HCF of variables

  • Both terms contain aa and bb
  • Lowest powers: a1a^1 and b1b^1

Step 3: HCF = 3ab3ab

Step 4: Factor out the HCF 3a2b+6ab2=3ab(a+2b)3a^2b + 6ab^2 = 3ab(a + 2b)

Check: 3ab×a=3a2b3ab \times a = 3a^2b ✓ and 3ab×2b=6ab23ab \times 2b = 6ab^2 ✓

Combining algebraic fractions

When adding or subtracting algebraic fractions with different denominators, you need to find a common denominator first. This is typically the lowest common multiple of all the denominators involved.

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Worked Example: Combining Algebraic Fractions

Simplify: x4t−2y5t+z2t\frac{x}{4t} - \frac{2y}{5t} + \frac{z}{2t}

Step 1: Find the LCM of denominators LCM of 4t4t, 5t5t, and 2t2t = 20t20t

Step 2: Convert each fraction to have denominator 20t20t

  • x4t=5x20t\frac{x}{4t} = \frac{5x}{20t}
  • 2y5t=8y20t\frac{2y}{5t} = \frac{8y}{20t}
  • z2t=10z20t\frac{z}{2t} = \frac{10z}{20t}

Step 3: Combine the fractions 5x20t−8y20t+10z20t=5x−8y+10z20t\frac{5x}{20t} - \frac{8y}{20t} + \frac{10z}{20t} = \frac{5x - 8y + 10z}{20t}

infoNote

When working with fractions, always check if your final answer can be simplified further by finding common factors in the numerator and denominator.

Common exam traps and tips

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Key Exam Tips:

  • Expanding brackets with negatives: One of the most frequent mistakes is forgetting to multiply all terms when expanding brackets, especially when there's a negative sign involved. Always double-check that you've applied the multiplier to every single term.

  • Finding the true HCF: When simplifying fractions, make sure you identify the complete highest common factor. Don't just focus on the numbers - consider all the variables and their powers too.

  • Factorising completely: Remember that you're looking for the largest expression that divides into all terms. Start with numbers, then look at variables, taking the lowest power of each variable that appears in every term.

  • Common denominators: When combining fractions, finding the correct common denominator is crucial. The lowest common multiple helps keep your working neat and reduces errors.

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Key Points to Remember:

  • Simplifying means writing expressions in their most compact form - this involves collecting like terms, expanding brackets, and reducing fractions where possible.

  • Like terms have identical variable parts - you can only combine terms that have exactly the same variables raised to the same powers.

  • When expanding brackets with negative signs, multiply ALL terms by the negative - this is the most common error in algebra, so always double-check your work.

  • Factorising requires finding the highest common factor - look at both the numerical coefficients and the variable parts to identify what can be taken out.

  • For algebraic fractions, find common denominators using the lowest common multiple - this makes adding and subtracting fractions much more manageable.

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