FACTORING ALGEBRAIC EXPRESSIONS Created by ﺟﻴﻄ for mathlabsky.wordpress.com Created by ﺟﻴﻄ for mathlabsky.wordpress.com

P Q P + Q P – Q P x Q FACTORING ALGEBRAIC EXPRESSIONS I. Arithmetical operations on algebraic : Adding Substracting Multiplying Dividing Exponentiating P Q P + Q P – Q P x Q 2x – 3 4x + 1 6x -2 -2x – 4 8x2 - 10x – 3 x + 2 2x – 1 3x + 1 -x + 3 2x2 + 3x – 2 2a – a2 2a 4a – a2 -a2 4a2 – 2a3 2(3 – a) 1 – a 7 – 3a 5 – a 6 – 8a + 2a2 4b - 1 (2b + 1) 6b 2b – 2 8b2 + 2b – 1

P Q P : Q P P2 P3 4a 2 2a 4a2 6a3 3a 2a2 12a – 4a2 6 – 2a 8a + 16 4 -3ab3 9a2b6 -27a3b9 4ab 16a2b2 64a3b3

II. Factoring Algebraic Expressions 12 ab2 – 9b3c2 = 3b2 (4a – 3bc2) 4x = 22.x 12ab2 = 22.3.a.b2 8 = 23 9b3c2 = 32.b3.c2 GCD = 22 = 4 GCD = 3.b2 Exercise 12xy2 + 4x2y3z 8x – 12y 24xyz2 + 9x2y 10xy3 + 2y2z 1. 4xy2 (3 + xyz) 2. 4 (2x - 3y) 3. 3xy (8z2 + 3x) 4. 2y2 (5xy + z)

III. Special form A. x2 + 2xy + y2 = (x + y) 2 B. x2 - 2xy + y2 = (x - y) 2 (x + y) 2 =(x + y)(x + y) (x - y) 2 =(x - y)(x - y) = x2 + xy + xy + y2 = x2 - xy - xy + y2 = x2 + 2xy + y2 = x2 - 2xy + y2 x2 + 4x + 4 = ( … + ...) 2 ( x + 2) 2 x2 - 6x + 9 = ( … - ...) 2 ( x - 3) 2

A. x2 + 2xy + y2 = (x + y) 2 B. x2 - 2xy + y2 = (x - y) 2 a2 + 4a + 4 Factor the following algebraic expressions! a2 + 4a + 4 16x2 – 24x + 9 4a2 – 4ab + b2 9m2 + 30mn + 25n2 25p2 + 70pq + 49q2 1. (a + 2)2 2. (4x - 3)2 3. (2a - b)2 4. (3m + 5n)2 5. (5p + 7q)2

a2 – b2 (2m) 2 – 32 x2 – 49 m2 – 121 64 – y2 1. (a – b)(a + b) C. x2 - y2 C. = (x - y)(x + y) (x – y )(x + y) = x2 + xy – xy - y2 = x2 - y2 a2 – b2 (2m) 2 – 32 x2 – 49 m2 – 121 64 – y2 1. (a – b)(a + b) 2. (2m – 3)(2m + 3) 3. (x – 7 )(x + 7) 4. (m – 11 )(m + 11) 5. (8 – y)(8 + y)

D. Factoring ax2 + bx + c, when a = 1 x2 + bx + c = (x + p)(x + q) Where c = p x q and b = p + q Example : 1. x2 + 10x + 16 ====> a = 1, b = 10, c = 16 p = …? q = …? … x … = 16 … + … = 10 8 2 8 2 x2 + 10x + 16  (x + 8)(x + 2) 2. x2 – x – 6 ====> a = 1, b = -1, c = -6 p = …? q = …? -3 … x … = -6 … + … = -1 2 x2 – x – 6  (x – 3)(x + 2) -3 2

a2 + 5a + 6 a2 + a – 6 (a + 3)(a + 2) (a – 2 )(a + 3) y2 + 6y + 9 Factor the following algebraic expressions! a2 + 5a + 6 a2 + a – 6 y2 + 6y + 9 y2 – 14y + 24 p2 + 4p – 5 (a + 3)(a + 2) (a – 2 )(a + 3) (y + 3)(y + 3) (y – 12)(y – 2) (p + 5)(p – 1)

E. Factoring ax2 + bx + c, when a ≠ 1 p + q = b p x q = a x c Example : 1. 2x2 + 7x + 6 ===> a = 2, b = 7, c = 6 p = …? q = …? … + … = 7 … x … = 12 4 3 4 3 2x2 + 7x + 6  2x2 + 4x + 3x + 6  2x2 + 4x + 3x + 6  2x(x + 2) + 3(x + 2)  (x + 2) (… + …) (2x + 3)

2. 6x2 + 13x - 5 ===> a = 6, b = 13, c = -5 p = …? q = …? … + … = 13 … x … = -30 -2 15 -2 15 6x2 + 13x - 5  6x2 - 2x + 15x - 5  6x2 - 2x + 15x - 5  2x(3x - 1) + 5(3x - 1)  (3x - 1) (… + …) (2x + 5)

IV. OPERATIONS ON ALGEBRAIC FRACTIONS Example: