1. Factorization of Simple Expressions
Factorization means writing an expression as a product of simpler expressions.
Common Factor Method: Take out the greatest common factor (GCF).
Example:
6x^2 + 9x
Step 1: Find GCF = 3x
Step 2: Factor out GCF → 3x*(2*x + 3)
2. Difference of Squares
Formula: a^2 - b^2 = (a + b)(a - b)
Example:
x^2 - 9
Step 1: Recognize 9 = 3^2
Step 2: Apply formula → (x + 3)(x - 3)
Example:
x^2 - 4y^2
Step 1: Recognize 4y^2 = (2y)^2
Step 2: Apply formula → (x + 2y)(x - 2*y)
3. Perfect Square Trinomials
Formulas:
a^2 + 2ab + b^2 = (a + b)^2
a^2 - 2ab + b^2 = (a - b)^2
Example:
x^2 + 6x + 9
Step 1: Recognize 6x = 2x3 and 9 = 3^2
Step 2: Apply formula → (x + 3)^2
Example:
y^2 - 10y + 25
Step 1: Recognize 10y = 2y5 and 25 = 5^2
Step 2: Apply formula → (y - 5)^2
4. Factorization of Quadratics
General quadratic: ax^2 + bx + c
Steps to factorize:
Find two numbers whose product = a*c and sum = b
Split the middle term using these numbers
Factor by grouping
Example:
x^2 - 5x + 6
Step 1: Find numbers → -2 and -3 (sum = -5, product = 6)
Step 2: Split middle term → x^2 - 2x - 3x + 6
Step 3: Group → (x^2 - 2x) - (3x - 6)
Step 4: Factor → x(x - 2) - 3*(x - 2)
Step 5: Final factorization → (x - 3)*(x - 2)
5. Sum and Difference of Cubes
Formulas:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Example:
x^3 - 8y^3
Step 1: Recognize 8y^3 = (2y)^3
Step 2: Apply formula → (x - 2y)(x^2 + 2xy + 4y^2)
Example:
x^3 + y^3
Step 1: Apply formula → (x + y)(x^2 - xy + y^2)
6. Factorization of Higher Powers
Some expressions like a^4 + a^2*b^2 + b^4 can be factorized using the formula:
a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)
Example:
x^4 + x^2 + 1
Step 1: Recognize pattern → (x^2)^2 + x^21 + 1^2
Step 2: Apply formula → (x^2 + x + 1)(x^2 - x + 1)
Example:
x^4 + 4y^4
Step 1: Recognize pattern → x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2*y^2)
7. Factorization by Grouping
Group terms in pairs and take common factors.
Example:
x^2 - 10x + 24 + 6y - 9y^2
Step 1: Group terms → (x^2 - 10x + 25) - (1 - 6y + 9y^2)
Step 2: Recognize squares → (x - 5)^2 - (1 - 3y)^2
Step 3: Apply difference of squares → (x - 3y - 4)(x + 3y - 6)
8. Special Cases
Complex expressions with multiple variables can often be factored using:
Difference of squares
Grouping
Sum/difference of cubes
Examples:
x^2 - 4y^2 = (x + 2y)(x - 2y)
x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)
9. Important Examples of Factorization
1.a^2 - b^2 = (a + b)*(a - b)
2.x^2 - 4y^2 = (x + 2y)(x - 2y)
3.z^2 - 10*z + 25 = (z - 5)^2
4.y^2 + 6*y + 9 = (y + 3)^2
5.a^2 - 2ab + b^2 = (a - b)^2
6.x^2 - 5x + 6 = (x - 2)(x - 3)
7.a^3 - b^3 = (a - b)(a^2 + ab + b^2)
8. x^3 - 8y^3 = (x - 2y)(x^2 + 2xy + 4y^2)
9. x^3 + y^3 = (x + y)(x^2 - xy + y^2)
10.8z^3 - 1 = (2z - 1)(4z^2 + 2*z + 1)
11.8a^3 + 36a^2b + 54ab^2 + 27 = (2a + 3*b)^3
10. Important Tips
Always check for common factors first.
For quadratics, try to split the middle term.
For higher powers, check difference/sum of squares/cubes formulas.
Use grouping carefully if stuck.
11. Important Questions (Step by Step)
Q1. Factorize x^4 + x^2 + 1
Step 1: Recognize pattern → (x^2)^2 + x^21 + 1^2
Step 2: Factor → (x^2 + x + 1)(x^2 - x + 1)
Q2. Factorize x^2 - 10x + 24 + 6y - 9*y^2
Step 1: Group → (x^2 - 10x + 25) - (1 - 6y + 9y^2)
Step 2: Recognize squares → (x - 5)^2 - (1 - 3y)^2
Step 3: Apply difference of squares → (x - 3y - 4)(x + 3*y - 6)
Q3. Factorize x^3 - 8*y^3
Step 1: Recognize 8y^3 = (2y)^3
Step 2: Apply formula → (x - 2y)(x^2 + 2xy + 4*y^2)
Q4. Factorize a^4 + a^2*b^2 + b^4
Step 1: Apply formula → (a^2 + ab + b^2)(a^2 - a*b + b^2)
Q5. Factorize x^3 + y^3
Step 1: Apply formula → (x + y)(x^2 - xy + y^2)
Q6. Factorize 8*z^3 - 1
Step 1: Recognize 8z^3 = (2z)^3
Step 2: Apply formula → (2z - 1)(4z^2 + 2z + 1)
Q7. Factorize x^2 - 4*y^2
Step 1: Apply difference of squares → (x + 2y)(x - 2*y)
Q8. Factorize z^2 - 10*z + 25
Step 1: Recognize perfect square → (z - 5)^2
Q9. Factorize y^2 + 6*y + 9
Step 1: Recognize perfect square → (y + 3)^2
Q10. Factorize x^2 - 5*x + 6
Step 1: Split middle term → x^2 - 2x - 3x + 6
Step 2: Group → (x^2 - 2x) - (3x - 6)
Step 3: Factor → x*(x - 2) - 3*(x - 2)
Step 4: Final → (x - 3)*(x - 2)
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