MathematicsChapter 4: Factorization and Algebraic Manipulation
Special Quartic Patterns
Some degree-4 expressions hide squares after a small trick.
Learning objectives
- Explain Special Quartic Patterns in simple mathematical language.
- Apply a^4 + a^2 b^2 + b^4 correctly.
- Apply a^4 + 4b^4 correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
Definition
Some degree-4 expressions hide squares after a small trick. The sum a^4 + a^2 b^2 + b^4 is not a perfect square as written, but adding and subtracting a^2 b^2 turns it into (a^2 + b^2)^2 - (ab)^2, a difference of squares. The related identity a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2). Both are standard Class 9 devices, not new operations.
Where / symbols
- a^4 + a^2 b^2 + b^4
- The sum a^4 + a^2 b^2 + b^4 is not a perfect square as written, but adding and subtracting a^2 b^2 turns it into (a^2 + b^2)^2 - (ab)^2, a difference of squares.
- adding
- The sum a^4 + a^2 b^2 + b^4 is not a perfect square as written, but adding and subtracting a^2 b^2 turns it into (a^2 + b^2)^2 - (ab)^2, a difference of squares.
- subtracting
- The sum a^4 + a^2 b^2 + b^4 is not a perfect square as written, but adding and subtracting a^2 b^2 turns it into (a^2 + b^2)^2 - (ab)^2, a difference of squares.
- a^2 b^2
- The sum a^4 + a^2 b^2 + b^4 is not a perfect square as written, but adding and subtracting a^2 b^2 turns it into (a^2 + b^2)^2 - (ab)^2, a difference of squares.
- a^4 + 4b^4
- The related identity a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2).
- (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2)
- The related identity a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2).
- a^2 - b^2
- Both tricks are 'add a square term you need, subtract it again, then use a^2 - b^2.
- 2a^2 b^2
- Write a^4 + 2a^2 b^2 + b^4 - a^2 b^2, that is add and subtract a^2 b^2.
- 2ab
- The related identity a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2).
Overview
Complete a square, then split as a^2 - b^2:
a^4 + a^2 b^2 + b^4
Start from a^4 + a^2 b^2 + b^4. Write a^4 + 2a^2 b^2 + b^4 - a^2 b^2, that is add and subtract a^2 b^2. Then a^4 + 2a^2 b^2 + b4 = (a2 + b2)2, so the expression is (a2 + b2)2 - (ab)2 = (a2 + b2 - ab)(a2 + b2 + ab). The extra a^2 b^2 you added is cancelled by the subtraction.
Examples x^4 + x^2 + 1 = (x^2 - x + 1)(x^2 + x + 1) · a^4 + a^2 b^2 + b^4 = (a^2 - ab + b^2)(a^2 + ab + b^2) · m^4 + m^2 n^2 + n^4 same pattern
a^4 + 4b^4
The identity a^4 + 4b^4 is rewritten as a^4 + 4a^2 b^2 + 4b^4 - 4a^2 b^2 = (a^2 + 2b^2)^2 - (2ab)^2. Factor as difference of squares: (a^2 + 2b^2 - 2ab)(a^2 + 2b^2 + 2ab), which is the same as (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2). The 4b^4 is (2b^2)^2, which is why the 4 is special.
Examples x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2) · a^4 + 4b^4 as in the identity · 16y^4 + 4 = 4(4y^4 + 1) then treat 4y^4 + 1 carefully
WHAT — Core idea
Some degree-4 expressions hide squares after a small trick. The sum a^4 + a^2 b^2 + b^4 is not a perfect square as written, but adding and subtracting a^2 b^2 turns it into (a^2 + b^2)^2 - (ab)^2, a difference of squares. The related identity a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2). Both are standard Class 9 devices, not new operations.
WHY — Why it matters
Both tricks are 'add a square term you need, subtract it again, then use a^2 - b^2.' If you only add and forget to subtract, the value changes and the factorisation is false.
HOW — Method and examples
1. a^4 + a^2 b^2 + b^4: Start from a^4 + a^2 b^2 + b^4. 2. a^4 + 4b^4: The identity a^4 + 4b^4 is rewritten as a^4 + 4a^2 b^2 + 4b^4 - 4a^2 b^2 = (a2 + 2b2)2 - (2ab)2.
HOW — Answer in an exam
a^4 + a^2 b^2 + b^4 is not (a^2 + b^2)^2; that would need 2a^2 b^2 in the middle. You must add and subtract a^2 b^2. Quote a^4 + 4b^4 with both trinomials, including the 2ab terms.
Explanation
Both tricks are 'add a square term you need, subtract it again, then use a^2 - b^2.' If you only add and forget to subtract, the value changes and the factorisation is false.
In simple words
a^4 + a^2 b^2 + b^4 is not (a^2 + b^2)^2; that would need 2a^2 b^2 in the middle. You must add and subtract a^2 b^2. Quote a^4 + 4b^4 with both trinomials, including the 2ab terms.
Key points
- a^4 + a^2 b^2 + b^4
- adding
- subtracting
- a^2 b^2
- add
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Special Quartic Patterns
Sticky Notes · Quick Revision
WHAT
- Some degree-4 expressions hide squares after a small trick.
Key Points
- Some degree-4 expressions hide squares after a small trick.
- Start from a^4 + a^2 b^2 + b^4.
- The identity a^4 + 4b^4 is rewritten as a^4 + 4a^2 b^2 + 4b^4 - 4a^2 b^2 = (a2 + 2b2)2 - (2ab)2.
- Both tricks are 'add a square term you need, subtract it again, then use a^2 - b^2.
HOW
- 1. a^4 + a^2 b^2 + b^4: Start from a^4 + a^2 b^2 + b^4.
- 2. a^4 + 4b^4: The identity a^4 + 4b^4 is rewritten as a^4 + 4a^2 b^2 + 4b^4 - 4a^2 b^2 = (a2 + 2b2)2 - (2ab)2.
Examples
- x4 + x2 + 1 = (x2 - x + 1)(x2 + x + 1)
- a^4 + a^2 b^2 + b^4 = (a2 - ab + b2)(a2 + ab + b2)
- m^4 + m^2 n^2 + n^4 same pattern
- x4 + 4 = (x2 + 2x + 2)(x2 - 2x + 2)
- a^4 + 4b^4 as in the identity
Common Mistake
- a^4 + a^2 b^2 + b^4 is not (a^2 + b^2)^2; that would need 2a^2 b^2 in the middle.
- You must add and subtract a^2 b^2.
Exam Tip
- a^4 + a^2 b^2 + b^4 is not (a^2 + b^2)^2; that would need 2a^2 b^2 in the middle. You must add and subtract a^2 b^2. Quote a^4 + 4b^4 with both trinomials, including the 2ab terms.
60-second revision
- Some degree-4 expressions hide squares after a small trick.
- Start from a^4 + a^2 b^2 + b^4.
- The identity a4 + 4b4 is rewritten as a4 + 4a2 b2 + 4b4 - 4a2 b2 = (a2 + 2b2)2 - (2ab)2.
- Both tricks are 'add a square term you need, subtract it again, then use a^2 - b^2.
- a^4 + a^2 b^2 + b^4 is not (a^2 + b^2)^2; that would need 2a^2 b^2 in the middle.
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