IB MYP 4-5 Maths- Factorizing Algebraic expressions - Study Notes - New Syllabus
IB MYP 4-5 Maths- Factorizing Algebraic expressions – Study Notes
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- Factorizing Algebraic expressions
IB MYP 4-5 Maths- Factorizing Algebraic expressions – Study Notes – All topics
Factorizing Algebraic Expressions
Factorizing Algebraic Expressions
To factorize means to write an algebraic expression as a product of its factors. It is the reverse of expanding brackets.
Why factorize?
- To simplify expressions
- To solve equations
- To understand structure and symmetry
How to Factorize – General Steps:
- Check for a common factor in all terms (take out GCF first if possible).
- Count the number of terms:
- 2 terms → Look for difference of squares or special forms like cube identities.
- 3 terms → Try factoring as a trinomial (quadratic form).
- 4 terms → Use grouping to split and factor in pairs.
- Check for special patterns like perfect square trinomials or sum/difference of cubes.
- Look for substitution if exponents are even or in the form of nested squares.
1. Taking out the Common Factor (GCF)
Look for the greatest common factor in all terms.
- \( 6x + 12 = 6(x + 2) \)
- \( 10ab – 15a = 5a(2b – 3) \)
Example:
Factorize: \( 8x^2 + 12x \)
▶️ Answer/Explanation
Common factor = 4x
\( 8x^2 + 12x = 4x(2x + 3) \)
Example:
Factorize: \( 14xy – 21xz \)
▶️ Answer/Explanation
Common factor = 7x
\( 14xy – 21xz = 7x(2y – 3z) \)
2. Factorizing by Grouping
Group terms in pairs and factorize each group.
- \( ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y) \)
- \( 3x^2 + 6x + 2x + 4 = 3x(x + 2) + 2(x + 2) = (3x + 2)(x + 2) \)
Example:
Factorize: \( ab + ac + xb + xc \)
▶️ Answer/Explanation
Group and factor:
\( ab + ac + xb + xc = a(b + c) + x(b + c) = (a + x)(b + c) \)
Example:
Factorize: \( 5m – 10n + 2m – 4n \)
▶️ Answer/Explanation
Group and factor:
\( (5m + 2m) + (-10n – 4n) = 7m – 14n = 7(m – 2n) \)
3. Factorizing Quadratic Trinomials
Case 1: Coefficient of \( x^2 \) is 1
- \( x^2 + 5x + 6 = (x + 2)(x + 3) \)
Example:
Factorize: \( x^2 + 7x + 12 \)
▶️ Answer/Explanation
Find two numbers that add to 7 and multiply to 12 → 3 & 4
\( (x + 3)(x + 4) \)
Example:
Factorize: \( x^2 – 9x + 20 \)
▶️ Answer/Explanation
Find two numbers that add to -9 and multiply to 20 → -4 & -5
\( (x – 4)(x – 5) \)
Case 2: Coefficient of \( x^2 \) not 1
Example:
Factorize: \( 2x^2 + 9x + 10 \)
▶️ Answer/Explanation
Find factors of 20 that sum to 9 → 4 & 5
\( 2x^2 + 4x + 5x + 10 = 2x(x + 2) + 5(x + 2) = (2x + 5)(x + 2) \)
Example:
Factorize: \( 3x^2 – 14x + 8 \)
▶️ Answer/Explanation
Multiply 3 × 8 = 24; Find factors of 24 that sum to -14 → -12 & -2
\( 3x^2 – 12x – 2x + 8 = 3x(x – 4) -2(x – 4) = (3x – 2)(x – 4) \)
4. Difference of Two Squares
- \( a^2 – b^2 = (a – b)(a + b) \)
Example:
Factorize: \( x^2 – 36 \)
▶️ Answer/Explanation
\( x^2 – 6^2 = (x – 6)(x + 6) \)
Example:
Factorize: \( 25a^2 – 49b^2 \)
▶️ Answer/Explanation
\( (5a)^2 – (7b)^2 = (5a – 7b)(5a + 7b) \)
5. Perfect Square Trinomials
Use these identities:
- \( a^2 + 2ab + b^2 = (a + b)^2 \)
- \( a^2 – 2ab + b^2 = (a – b)^2 \)
- \( x^2 + 6x + 9 = (x + 3)^2 \)
- \( x^2 – 8x + 16 = (x – 4)^2 \)
Example:
Factorize: \( x^2 + 10x + 25 \)
▶️ Answer/Explanation
\( x^2 + 10x + 25 = x^2 + 2(5)x + 5^2 = (x + 5)^2 \)
Example:
Factorize: \( 4x^2 – 12x + 9 \)
▶️ Answer/Explanation
\( 4x^2 – 12x + 9 = (2x)^2 – 2(2x)(3) + 3^2 = (2x – 3)^2 \)
6. Factorizing Cubic Expressions
Case 1: Factor by Grouping
- \( x^3 + 3x^2 + 2x + 6 = x^2(x + 3) + 2(x + 3) = (x^2 + 2)(x + 3) \)
Case 2: Special Identities
- \( a^3 + b^3 = (a + b)(a^2 – ab + b^2) \)
- \( a^3 – b^3 = (a – b)(a^2 + ab + b^2) \)
Example:
Factorize: \( x^3 + 27 \)
▶️ Answer/Explanation
\( x^3 + 27 = x^3 + 3^3 = (x + 3)(x^2 – 3x + 9) \)
Example:
Factorize: \( 8x^3 – 125 \)
▶️ Answer/Explanation
\( 8x^3 – 125 = (2x)^3 – 5^3 = (2x – 5)(4x^2 + 10x + 25) \)
7. Miscellaneous Techniques
Case 1: Factoring out negatives
- \( -x^2 + 4x = -1(x^2 – 4x) \)
Case 2: Substitution
- \( x^4 + 5x^2 + 6 = y^2 + 5y + 6 \) where \( y = x^2 \), so factor as \( (x^2 + 2)(x^2 + 3) \)
Example:
Factorize: \( -2x^2 + 10x \)
▶️ Answer/Explanation
Factor out the negative:
\( -2x^2 + 10x = -2x(x – 5) \)
Example:
Factorize: \( x^4 – 13x^2 + 36 \)
▶️ Answer/Explanation
Let \( y = x^2 \). Then: \( y^2 – 13y + 36 \)
\( = (y – 4)(y – 9) \Rightarrow (x^2 – 4)(x^2 – 9) \)
Now apply difference of squares:
\( = (x – 2)(x + 2)(x – 3)(x + 3) \)