Comprehensive Guide to Factorization in Mathematics
Explore the concept of factorization in algebraic expressions, including the definition of factors, common factors, methods of factorization, and illustrated examples. Learn about finding factors, common factors, and various factorization techniques such as common factors method and regrouping of terms method. Enhance your understanding of factorization through clear explanations and step-by-step solutions.
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T TEACH A A NEED EACH M NEED BASED MAYURBHANJ AYURBHANJ BASED EDUCATIONAL EDUCATIONAL DRIVE DRIVE Subject: Mathematics Topic: Factorisation M r. Sagar K um ar M r. Sagar K um ar D alnaik D alnaik Int. Int. B .E d B .E d- -M .E d M .E d. 1 . 1 st stYear N orth Orissa U niversity B aripada N orth Orissa U niversity B aripada Year
FACTORIZATION Factorization of algebraic expressions: The process of breaking a number of algebraic expression into a product of many factors of other numbers or terms, when we multiplied it gives the original number is called factorization. Factorization of algebraic expression means to express any algebraic expression as the product of two or more algebraic expressions. Factorization of any Algebraic expression can be uniquely express as the product of its factors apart from order of it s the factors. The factorization formula divides a large number or term into small number or term, is known as factors.
What is a Factor? Factor are the numbers, algebraic variables or an algebraic expression which divides the number or an algebraic expression without leaving remainder. For example factor of 9 is 1, 3 and 9. Find the factor of numbers 45, 75, and 210. Ans. 45 = 3 X 3X 5, factor of 9 is 1, 3 and 5. 75 = 3 X 5 X 5 , factor of 9 is 1, 3 and 5. 210 = 2 X 3 X 5 X 7, factor of 9 is 1,2, 3,5 and 7. Note. 1 is factor of every integer. Common Factor: In an algebraic expression or number the term which is common to both numbers or algebraic expression, is known as common factor. For example common factors of 9 & 15 are 1 and 3. Find the common factors of 52 & 104y? Ans. 52 = 52 x 1, 104y = 1 x 52 x 2 x y So, common factors are 1 & 52.
ILLUSTRATIONOF FACTORIZATION Factorize 26xy-169y. Ans. 26xy-169y= 13.y(2x-13) Some examples are given 24x3 + 16x2 + 4x = 2. 2. x (6x2 + 4x + 1) 2px 2py + 3qx 3qy = 2.p (x y)+3.q (x y) =(x y) ( 2p + 3q) 21y 35 x+9y2 15xy =7(3y 5x)+3y(3y 5x) = (3y 5x) (7+3y) Methods of Factorization: (a) Common Factors methods (b) Regrouping of terms methods Factorization using algebraic identities (d) Factorization by splitting the middle terms (c)
(A) COMMON FACTORS METHOD In this method, we simply take out the common factors among each term of the given expression. Example, Factorize 3x +9 Since, 3 is the common factor for both the terms 3x and 9, thus taking 3 as a common factor we get 3x +9 = 3(x+3) More Example are given Example1. Factorize 25xy-160py Ans. 25xy-160py =5.y (5x-32p) Example 2. Factorize 21a2b +49ab2 Ans. 21a2b +49ab2 =7ab (3a+7b) Example 3. 25ab -210bcd = 5.b (5a 42cd) Example 4. 81xyz 147pqx = 3.x (27yz 49pq)
(B) REGROUPINGOFTERMS METHOD Regrouping of terms method means to rearrange the given expression based on the like terms or similar terms. For example, 2xy +3+2y +3x can be rearranged as 2xy +3x +2y+3, and proceeds further to factorizing the terms as follows. 2xy +3x+2y +3= 2xy +3x +2y+3 = 2.x.y +3.x +2.y+3 = x (2y+3)+1 (2y+3) =(2y+3)(x+1) Example : Factorize 7pq+21q+p+3 Ans. 7pq+21q+p+3= 7q(p+3)+1(p+3) =(p+3)(7q+1)
(C) FACTORIZATIONUSING ALGEBRAIC IDENTITIES Algebraic Identity: The algebraic equations which are true for all values of variables in them is called algebraic identities. Factorization by Method of Using Algebraic identities : (a + b)2 = a2+2ab+b2 (a b)2 = a2-2ab +b2 a2 b2 = (a+b)(a b) (a + b)3 = a3 + 3a2b + 3ab2 + b3 = a3 + 3ab(a + b) + b3 (a b)3 = a3 -3a2b + 3ab2 - b3 = a3 3ab( a b) b3 a3 + b3 = (a + b)(a2 ab + b2) a3 b3 = (a b)(a2 + ab + b2) a4 b4 = (a-b)(a+b)(a2 +b2 ) (a + b+c)2 = a2 +b2 +c2 +2ab + 2bc + 2ca (a -b-c)2 = a2 +b2 +c2 - 2ab + 2bc - 2ca
ILLUSTRATIONOF FACTORIZATION USING ALGEBRAIC IDENTITIES Example 1: Factorize x2+10x+25 Ans. x2+10x+25=(x)2+2.x.5+(5)2=(x+5)2 =(x + 5)(x + 5) Example 2: Factorize a2 6ab+9b2 Ans. a2 6ab+9b2 = (a)2 2.a.3b+(3b)2 =(a 3b)2 = (a 3b) (a 3b) Example 3: Factorize a2 225 Ans. a2 225=(a)2 (15)2= (a + 15)(a 15) Example 4: Factorize a3 125 Ans. a3 125=(a)3 (5)3 = (a-5) (a2 +5a+25)
(D) FACTORIZATIONBYSPLITTINGTHE MIDDLETERM Factorization by using the following formulae (1) (x + a)(x + b) = x2+(a+b)x+b2 (2) (x - a)(x - b) = x2 -(a+b)x+b2 1. Factorize : x2 + 5x + 6 3. Factorize: 32x2 + 48x + 18 = x2 + (2 + 3 )x + 2x3 = 2(16x2 + 24x + 9) = x2 + 2x + 3x + 6 = 2[(4x)2 + 2.4x.3 + (3)2] = x(x + 2) + 3(x + 2) = 2(4x + 3)2 = (x + 2)(x + 3) = 2 (4x + 3)(4x + 3) 2. Factorize: 2x2 7x + 6 = 2x2 ( 4 + 3 )x + 6 = 2x2 4x 3x + 6 = 2x(x 2 ) 3(x 2) = ( 2x 3 )(x 2 )
ILLUSTRATIONOF FACTORIZATIONBY SPLITTINGTHEMIDDLETERM Example 1. Factorize 6x2 + 17x + 12 Ans. 6x2 + 17x + 12 = 6x2 + 9x + 8x + 12 = 3x(2x + 3) + 4(2x + 3) = (2x + 3)(3x + 4) Example 2. Factorize x2 - 13x - 140 Ans. x2-13x -140 = x2 (20 7)x - 140 = x2 20x + 7x - 140 = x(x - 20) + 7(x - 20) = (x - 20)(x + 7)
TEST ZONE Factorize the following x2+30x+225 p2+10p+25 x3 -1331 x3 +729 a4 625 a2 - a+2 2a2 - 8a+8 a2 +10a+21 2x2+40x+25 a3 +21 a2 +147a +343 4 a2 +9 b2 +16 c2 -12ab+24bc 16ca 25 a2 + b2 +36 c2 +10ab+12bc + 60ca
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