CIE IGCSE Mathematics (0580) Algebraic manipulation Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Algebraic manipulation Study Notes
LEARNING OBJECTIVE
- Algebraic Manipulation
Key Concepts:
- Simplify expressions by collecting like terms.
- Expand products of algebraic expressions.
- Factorise by extracting common factors.
Simplifying Expressions by Collecting Like Terms
1. Simplifying Expressions by Collecting Like Terms
In algebra, like terms are terms that have the same variables raised to the same powers. You can only simplify expressions by combining like terms — that means adding or subtracting their coefficients.
For example, \( 3x \) and \( 5x \) are like terms, but \( 3x \) and \( 5x^2 \) are not.
Steps to Simplify:
- Group like terms together.
- Add or subtract the coefficients of those like terms.
- Leave unlike terms as they are.
Example:
Simplify: \( 3x + 4y – 2x + 6y \)
▶️ Answer/Explanation
Group like terms: \( (3x – 2x) + (4y + 6y) \)
Simplify: \( x + 10y \)
Answer: \( x + 10y \)
Example:
Simplify: \( 4x + 7 – 2x + 5 – 3x \)
▶️ Answer/Explanation
Group like terms:
\( (4x – 2x – 3x) + (7 + 5) \)
\( = -1x + 12 \)
Answer: \( -x + 12 \)
Expanding Products of Algebraic Expressions
2. Expanding Products of Algebraic Expressions
Expanding an algebraic expression means removing brackets by multiplying each term inside the bracket by the term outside (or by each term of the other bracket, if binomial).
Common Types of Expansions:
- Single bracket: Multiply each term inside by the factor outside.
e.g. \( 3(x + 5) = 3x + 15 \) - Double brackets (binomials): Use distributive property (FOIL – First, Outside, Inside, Last).
e.g. \( (x + 2)(x + 4) = x^2 + 6x + 8 \)
Tip:
Always multiply carefully and combine like terms after expanding.
Example
Expand: \( 5(2x – 3) \)
▶️ Answer/Explanation
\( 5 \times 2x = 10x \), \( 5 \times (-3) = -15 \)
Answer: \( 10x – 15 \)
Example
Expand: \( (x + 3)(x + 7) \)
▶️ Answer/Explanation
\( x \times x = x^2 \)
\( x \times 7 = 7x \)
\( 3 \times x = 3x \)
\( 3 \times 7 = 21 \)
Now combine: \( x^2 + 7x + 3x + 21 = x^2 + 10x + 21 \)
Answer: \( x^2 + 10x + 21 \)
Factorising by Extracting Common Factors
3. Factorising by Extracting Common Factors
Factorising is the reverse of expanding. It means writing an expression as a product of its factors. The first step in factorising is to look for common factors in all terms of the expression.
Steps to Factorise:
- Identify the highest common factor (HCF) of all terms.
- Take the HCF outside the bracket.
- Divide each term by the HCF and write the results inside the bracket.
Tip:
Always check your answer by expanding to ensure it’s equivalent to the original expression.
Example :
Factorise: \( 6x + 9 \)
▶️ Answer/Explanation
Common factor of 6x and 9 is 3.
\( 6x + 9 = 3(2x + 3) \)
Answer: \( 3(2x + 3) \)
Example :
Factorise: \( 4x^2 – 10x \)
▶️ Answer/Explanation
Common factor is 2x.
\( 4x^2 – 10x = 2x(2x – 5) \)
Answer: \( 2x(2x – 5) \)
Factorising Algebraic Expressions
Factorising Algebraic Expressions
Factorising means expressing an algebraic expression as a product of its factors. It is the reverse of expanding brackets and is a key skill for simplifying, solving equations, and understanding algebraic structure.
1. Common Factor: Expressions like \( ax + bx + kay + kby \)
Group terms and factor common elements from each group.
Example:
Factorise: \( 3x + 6y + 2ax + 4ay \)
▶️ Answer/Explanation
Group terms: \( (3x + 6y) + (2ax + 4ay) \)
Factor each group: \( 3(x + 2y) + 2a(x + 2y) \)
Now factor common binomial: \( (x + 2y)(3 + 2a) \)
2. Difference of Squares: \( a^2x^2 – b^2y^2 \)
Use the identity \( A^2 – B^2 = (A – B)(A + B) \).
Example:
Factorise: \( 9x^2 – 16y^2 \)
▶️ Answer/Explanation
\( = (3x)^2 – (4y)^2 \)
\( = (3x – 4y)(3x + 4y) \)
3. Perfect Square Trinomial: \( a^2 + 2ab + b^2 \)
Use the identity \( a^2 + 2ab + b^2 = (a + b)^2 \)
Example:
Factorise: \( x^2 + 6x + 9 \)
▶️ Answer/Explanation
\( = (x + 3)^2 \)
Since \( 6 = 2 \cdot x \cdot 3 \) and \( 9 = 3^2 \)
4. Quadratic Trinomial: \( ax^2 + bx + c \)
Look for two numbers whose product is \( a \times c \) and sum is \( b \). Then factor by grouping.
Example:
Factorise: \( 2x^2 + 5x + 3 \)
▶️ Answer/Explanation
Find two numbers that multiply to \( 2 \times 3 = 6 \) and add to 5: (2, 3)
\( 2x^2 + 2x + 3x + 3 \)
\( = 2x(x + 1) + 3(x + 1) = (x + 1)(2x + 3) \)
5. Common Factor in Cubic Expression: \( ax^3 + bx^2 + cx \)
First factor out the common term (usually \( x \)), then factor the remaining quadratic if possible.
Example:
Factorise: \( x^3 – 6x^2 + 9x \)
▶️ Answer/Explanation
Take out common factor: \( x(x^2 – 6x + 9) \)
Factor quadratic: \( x(x – 3)^2 \)
Completing the Square
Completing the Square
Completing the square is a method used to write a quadratic expression of the form \( ax^2 + bx + c \) in the form \( a(x + p)^2 + q \). This is helpful for solving equations, finding turning points of parabolas, and working with transformations.
General Process:
- If \( a \ne 1 \), factor it out from the quadratic and linear terms.
- Complete the square inside the bracket by adding and subtracting the square of half the coefficient of \( x \).
- Simplify and write in the form \( a(x + p)^2 + q \).
Example:
Write \( x^2 + 6x + 5 \) in completed square form.
▶️ Answer/Explanation
Step 1: Take half of 6 → \( \frac{6}{2} = 3 \)
Step 2: Square it → \( 3^2 = 9 \)
Step 3: Add and subtract 9: \( x^2 + 6x + 9 – 9 + 5 \)
\( = (x + 3)^2 – 4 \)
Final answer: \( (x + 3)^2 – 4 \)
Example:
Write \( 2x^2 + 8x + 3 \) in completed square form.
▶️ Answer/Explanation
Step 1: Factor out 2 → \( 2(x^2 + 4x) + 3 \)
Step 2: Half of 4 is 2, and \( 2^2 = 4 \)
Step 3: \( = 2[(x + 2)^2 – 4] + 3 \)
Step 4: Expand → \( 2(x + 2)^2 – 8 + 3 = 2(x + 2)^2 – 5 \)
Final answer: \( 2(x + 2)^2 – 5 \)
Example:
Write \( -3x^2 + 12x – 1 \) in completed square form.
▶️ Answer/Explanation
Step 1: Factor out -3 → \( -3(x^2 – 4x) – 1 \)
Step 2: Half of -4 is -2, and \( (-2)^2 = 4 \)
Step 3: \( = -3[(x – 2)^2 – 4] – 1 \)
Step 4: Expand → \( -3(x – 2)^2 + 12 – 1 = -3(x – 2)^2 + 11 \)
Final answer: \( -3(x – 2)^2 + 11 \)