1. factorization

2. We know that,
Factors of 12
2, 3, 4, 6.
Prime factors of

3. Further factors of Factor 1 Further factors of Factor 2
What is factorization.?
The process of expressing any polynomial as a product of it’s factors is called FACTORIZATION.
Eg. 9x² = (3x) (3x)
2a²b + 2ab² = (2ab) (a + b)
3x² + x -6 = (3x - 3) (x + 2)
Algebraic expression
Factor 1
Factor 2
Further factors of Factor 1
Further factors of Factor 2
9x²
(3x)
3x = 3 . x
2a²b + 2ab²
(2ab)
(a + b)
2ab = 2 . a. b
3x² + x – 6
(3x - 3)
(x + 2)

4. Points to remember,
A whole number greater than 1 for which the only factors are 1 and itself, is called a prime number. Eg. 2,3,4,5…
A whole number greater than 1 which has more than 2 factors is called a composite number.
Eg. 4,6,8,10…
1 is a factor of any number, 1 is neither prime nor composite.
Every natural number other than 1 is neither prime nor composite.
2 is the only even prime number.

5. Factorization by Taking out the common factor
Eg x + 8 = 4x + (2 . 4) = 4 (x + 2)
4 is common to both terms.
The factors of 4x + 8 = 4 and (x + 2)
Factors 4 and (x + 2) are irreducible
[Note: a factors that can’t be factorized further is known as irreducible factors]

6. Factorization by Grouping the terms…
Eg.
X³ - 2x² + x – 2 = x²(x -2) + 1(x - 2)
[step 1: In first two terms x² is common - In second term 1 is common - lets take out those 2 term]
= (x² + 1) (x – 2)
[step 2: let’s group the common term from the both first and
second term (i.e) (x – 2),
then group the remaining terms in another group]

7. Factorization by using identities… “polynomial expression can be Recall, factorized by using IDENTITIES..”
(x + y)²
(x + y) (x + y)
x² + 2xy +y²
(x - y)²
(x - y) (x - y)
x² - 2xy +y²
x² - y²
(x + y) (x - y) -
(x + a) (x + b)
X2 + (a+b)x + ab

8. Examples… Factorize the following expression by using identities.
1. x² + 4x + 4
Solution,
x² + 4x + 4 = x² + 2(2)x [compare with suitable Identity]
= (x + 2)²
= (x + 2) (x + 2)
i.e, (a + b)² = a² + 2ab + b²
= x² + 4x a = x, 2ab = 2x(2), b = 2

9. 2. x² + 4x + 4 Solution, x² + 6x + 9 = x² - 2x(3) + 9 [compare with
2. x² + 4x + 4 Solution, x² + 6x + 9 = x² - 2x(3) + 9 [compare with suitable Identity] = (x - 3)² = (x - 3) (x - 3)
i.e., (a-b)² = a² - 2ab + b²
= x² - 2x(3) a = x, 2ab = 2x(3), b = 3

10. 3. m4 – n4 solution: m4 – n4 = m4 – n [compare with suitable Identity (a2 – b 2)] = (m2)2 – (n 2) = (m2 + n2) (m2 – n2) = (m2 + n2) (m + n) (m – n)
i.e, (a2 – b2) = (a - b) (a + b)
m4 – n4
a2 = (m2)2 a = m2
b2 = (n2)2 b = n2

11. x2+(a + b)x + ab = (x + a) (x + b)
4. x² + 7x solution: x² + 7x + 12 = x² + 7x [compare with suitable Identity] = x² + (3 + 4)x = x² + 3x + 4x = (x² + 3x) + (4x + 12) = (x.x + 3.x) + (4.x +3.4) = x (x + 3) + 4 (x + 3) = (x + 3) (x + 4)
x2+(a + b)x + ab = (x + a) (x + b)
x² + 7x + 12
a + b = 7, ab = 12 a b
a + b ab
choice 1 6 7 2 5 10 3 4 12

12. x2+ (a + b)x + ab = (x + a) (x + b)
5. x² - 3x - 4 solution: x² - 3x - 4 = x² - 3x [compare with suitable identity] = x² + (1 + (-4))x + (1.(-4)) = x² + (1x + (-4)x) + (1.(-4)) = (x² + 1x) + ((-4)x + 1.(-4)) = (x.x + 1.x) + ((-4).x +1.(-4)) = x (x + 1) + (-4) (x + 1) = (x + 1) (x + (- 4)) = (x + 1) (x - 4)
x2+ (a + b)x + ab = (x + a) (x + b)
x² - 3x - 4
a + b = -3, ab = -4 a b
a + b ab
choice -1 3 2 -3 -2 -4 1

13. 6. 4x²y - 7xy². Solution:. 4x²y - 7xy² = 4x²y - 7xy²
6. 4x²y - 7xy² Solution: 4x²y - 7xy² = 4x²y - 7xy² [separate the terms] = (4.x.x.y) – (7.x.y.y) [take out the common terms x.y] = xy (4x – 7y) [group the terms]

14. Try These Factorize the following: y2− 8y + 2y – 16 x2 +7x + 10