Welcome to Algebraic Manipulation! Your Key to Solving Maths Puzzles
Hello future mathematician! Don't worry if algebra sometimes feels like a foreign language. This chapter, Algebraic Manipulation, is all about learning the essential rules for tidying up mathematical expressions and rearranging them—like organising your messy desk so you can find what you need quickly!
Mastering these techniques is absolutely crucial because they are the foundation for everything else you will study in the "Equations, formulae and identities" section and beyond. Let's dive in!
Quick Review Box: The Language of Algebra
- An Expression is a combination of terms (e.g., \(3x + 5\)).
- A Term is a single number, a variable, or numbers and variables multiplied together (e.g., \(3x\), \(5\), \(y^2\)).
- An Equation has an equals sign (\(=\)) and states that two expressions are equal (e.g., \(3x + 5 = 14\)).
Section 1: Substitution and Simplifying Expressions
1.1 Substitution: Giving Variables a Numerical Value
Substitution means replacing a variable (like \(x\) or \(y\)) with a specific number so you can calculate the value of the expression. Think of the variable as an actor, and the number is the role they are playing for the day!
Step-by-Step: Evaluating Expressions
Let's find the value of the expression \(4a - b^2\) when \(a = 3\) and \(b = -2\).
- Replace the variables: Always use brackets when substituting negative numbers to avoid sign errors!
\(4(3) - (-2)^2\) - Apply BIDMAS/BODMAS: Handle powers first.
Remember: \((-2)^2 = (-2) \times (-2) = +4\).
\(4(3) - 4\) - Perform multiplication:
\(12 - 4\) - Calculate the final answer:
\(8\)
Common Mistake to Avoid: Squaring a negative number! Many students write \((-2)^2 = -4\). This is incorrect. A negative number multiplied by a negative number is always positive.
1.2 Simplifying by Collecting Like Terms
Simplifying means making an expression look neater by combining terms that are mathematically the same.
Key Concept: Like terms must have the exact same variable(s) raised to the exact same power(s).
Analogy: Imagine you are organising fruit. You can easily add apples to apples, and bananas to bananas, but you cannot add apples and bananas together to get "apple-bananas"!
- \(3a\) and \(-5a\) are like terms (both have \(a^1\)).
- \(4x^2\) and \(x^2\) are like terms (both have \(x^2\)).
- \(2x\) and \(2x^2\) are NOT like terms.
Example: Simplifying \(5x + 3y - x + 7xy - 2y\)
- Identify like terms: Group them mentally (or physically underline them).
x-terms: \(5x\) and \(-x\)
y-terms: \(3y\) and \(-2y\)
xy-term: \(7xy\) (no partner) - Combine the groups: Always keep the sign immediately in front of the term.
\((5x - x) + (3y - 2y) + 7xy\) - Write the final simplified expression:
\(4x + y + 7xy\)
Key Takeaway for Section 1: Treat variables as quantities you can only combine if they are identical (like terms). Always check the sign in front of the term!
Section 2: Expanding Brackets
Expanding is the process of multiplying out terms enclosed in brackets. It follows the Distributive Law: the term outside the bracket must be multiplied by every single term inside.
2.1 Expanding Single Brackets
Step-by-Step Example: \(4(2x - 5)\)
- Multiply the outside term (4) by the first term inside (\(2x\)):
\(4 \times 2x = 8x\) - Multiply the outside term (4) by the second term inside (\(-5\)):
\(4 \times (-5) = -20\) - Combine the results:
\(8x - 20\)
Example with Variables Outside: \(3a(2a + 4b)\)
- \(3a \times 2a = 6a^2\) (Remember: \(a \times a = a^2\))
- \(3a \times 4b = 12ab\)
- Result: \(6a^2 + 12ab\)
2.2 Expanding Double Brackets (Binomials)
When you multiply two sets of brackets, like \((x+3)(x-2)\), you must make sure that every term in the first bracket is multiplied by every term in the second bracket.
We use the FOIL method as a great memory aid!
The FOIL Mnemonic
- First: Multiply the first terms in each bracket.
- Outer: Multiply the two outermost terms.
- Inner: Multiply the two innermost terms.
- Last: Multiply the last terms in each bracket.
Step-by-Step Example: \((x + 5)(x - 3)\)
\( (x + 5)(x - 3) \)
- First: \(x \times x = x^2\)
- Outer: \(x \times (-3) = -3x\)
- Inner: \(5 \times x = +5x\)
- Last: \(5 \times (-3) = -15\)
- Simplify (Collect the like terms in the middle):
\(x^2 - 3x + 5x - 15\)
\(x^2 + 2x - 15\)
Crucial Case: The Perfect Square
When you square a bracket, do not just square the terms inside!
\((x + 4)^2\) is NOT equal to \(x^2 + 16\).
You must rewrite it as two sets of brackets and use FOIL:
\((x + 4)^2 = (x + 4)(x + 4)\)
\( = x^2 + 4x + 4x + 16 \)
\( = x^2 + 8x + 16 \)
Key Takeaway for Section 2: Expanding uses the distributive law. For double brackets, FOIL ensures you multiply every pair, and always simplify the resulting expression by collecting the middle terms.
Section 3: Factorising Expressions (Reversing Expansion)
Factorising is the opposite of expanding. Instead of removing brackets, we are putting them back in! This process usually involves turning an expression (a sum of terms) into a product (terms multiplied together).
3.1 Factorising by Finding the Common Factor
This is the most common and simplest type of factorisation. We look for the Highest Common Factor (HCF) in both the numbers and the variables.
Step-by-Step Example: Factorise \(12x + 18\)
- Find the HCF of the numbers (12 and 18): The largest number that divides into both is 6.
- Find the HCF of the variables: There is an \(x\) in the first term, but not the second. So, there is no common variable factor.
- Place the HCF outside the bracket:
\(6(\space \space \space \space)\) - Divide each original term by the HCF to find the contents of the bracket:
\(12x \div 6 = 2x\)
\(18 \div 6 = 3\) - Final Factorised form:
\(6(2x + 3)\)
Example with Variable Factors: Factorise \(10a^2b - 15ab\)
- HCF of numbers (10 and 15): 5.
- HCF of variables: Both terms have at least \(a\) and at least \(b\). So the common variable factor is \(ab\).
- Overall HCF: \(5ab\)
- Divide the original terms by \(5ab\):
\((10a^2b) \div (5ab) = 2a\)
\((-15ab) \div (5ab) = -3\) - Final form:
\(5ab(2a - 3)\)
3.2 Factorising the Difference of Two Squares (DOTS)
This is a very specific, but very quick, factorisation technique. You MUST recognise the pattern!
An expression can be factorised using DOTS if it meets three criteria:
- It must have only two terms.
- The operation must be subtraction (a difference).
- Both terms must be perfect squares (e.g., \(x^2\), \(9\), \(4y^2\), \(100\)).
The general formula is:
\( \mathbf{a^2 - b^2 = (a - b)(a + b)} \)
Example: Factorise \(x^2 - 49\)
- Identify \(a^2\) and \(b^2\):
\(a^2 = x^2\), so \(a = x\)
\(b^2 = 49\), so \(b = 7\) - Apply the formula:
\((x - 7)(x + 7)\)
Example: Factorise \(25y^2 - 1\)
- \(a^2 = 25y^2\), so \(a = 5y\)
- \(b^2 = 1\), so \(b = 1\)
- Result: \((5y - 1)(5y + 1)\)
3.3 Factorising Simple Quadratic Expressions (\(x^2 + bx + c\))
A quadratic expression has an \(x^2\) term. To factorise simple quadratics (where the coefficient of \(x^2\) is 1), we look for two numbers that satisfy two conditions:
- They must MULTIPLY to equal the constant term (\(c\)).
- They must ADD to equal the coefficient of \(x\) (\(b\)).
Step-by-Step Example: Factorise \(x^2 + 7x + 10\)
We need two numbers that multiply to 10 and add to 7.
- List factors of 10:
(1 and 10), (2 and 5), (-1 and -10), (-2 and -5) - Check which pair adds up to 7:
\(2 + 5 = 7\). This is the correct pair! - Write the answer in two brackets:
\((x + 2)(x + 5)\)
Example with Negative Terms: Factorise \(x^2 - 3x - 18\)
We need two numbers that multiply to \(-18\) and add to \(-3\).
- Since they multiply to a negative number, one factor must be positive and one must be negative.
- Factors of 18 (looking for a difference of 3): (1, 18), (2, 9), (3, 6).
- The pair (3 and 6) has a difference of 3. Since the sum must be \(-3\), the larger number (6) must be negative.
\(+3\) and \(-6\) (Check: \(3 \times -6 = -18\); \(3 + (-6) = -3\). Correct!) - Final Factorised form:
\((x + 3)(x - 6)\)
Key Takeaway for Section 3: Factorising is reversing FOIL. Always look for a common factor first. Recognise DOTS instantly! For quadratics, the puzzle is finding the pair that multiplies to the end number and adds to the middle number.
Section 4: Review of Index Laws in Manipulation
When manipulating algebraic expressions, especially during expansion or simplification, you must correctly apply the rules of indices (powers).
The Three Essential Rules for Algebraic Manipulation
1. Multiplication Rule (Adding Powers)
When multiplying terms with the same base, add the exponents:
\( a^m \times a^n = a^{m+n} \)
Example: \( 5x^3 \times 2x^4 = (5 \times 2) \times (x^{3+4}) = 10x^7 \)
2. Division Rule (Subtracting Powers)
When dividing terms with the same base, subtract the exponents:
\( a^m \div a^n = a^{m-n} \)
Example: \( \frac{12y^5}{4y^2} = (12 \div 4) \times (y^{5-2}) = 3y^3 \)
3. Power of a Power Rule (Multiplying Powers)
When raising a power to another power, multiply the exponents:
\( (a^m)^n = a^{mn} \)
Example: \( (3x^2)^3 = 3^3 \times (x^2)^3 = 27x^6 \)
Did you know? Algebraic manipulation techniques were developed extensively in the 9th century by the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī—the word 'algebra' comes from the title of his famous book!
Summary and Encouragement
You have covered the fundamental tools of algebraic manipulation! Remember that mathematics is built layer by layer. If you struggle with factorising, go back and practice expanding, because they are reverse processes.
- Substitution: Replace variables carefully, using BIDMAS/BODMAS.
- Simplifying: Only combine like terms (same variable, same power).
- Expanding: Multiply everything inside the bracket by everything outside (use FOIL for double brackets).
- Factorising: Look for HCF, then check for DOTS or quadratic patterns.
Keep practising these skills! The more you use them, the more natural they will become. You've got this!