Mathematics department Algebra Unit one Factorization of a trinomial with the form ax 2 ± bx + c a=1of Factorization of a trinomial with the form ax 2 ± bx + c a ≠ 1 Factorization of a trinomial with the form ax 2 ± bx – c a = 1 Factorization of a trinomial with the form ax 2 ± bx – c a ≠ 1 Factorization of a perfect square By Mr. Mahmoud Sultan

(I) Factorization by getting H.C.F a) 10x 3 + 5x x= 5x (2x 2 + x + 12) b) 5x + 5y – 5z = 5 ………………..

Lesson (1) Factorization of a trinomial with the form ax 2 ± bx + c a=1 case one if the sign of the last term is positive is (+ve): Ex (1): Factorize: X 2 + 5X +6 Step 1: open two brackets and put ( half power of first term ) x in each (x ) (x ) solution Step 2: the sign in the brackets are the same sign of the middle term ( x + ) ( x + ) Step 3: search about two numbers the product of them (ac) is 6 and their sum (b) is 5 2 and 3 X 2 + 5X +6 = (x+2) (x+3)

1)Try by your self (on 12 min. ) Factorize the following four question :- 1]x 2 - 5x + 6 2]x 2 + 5x + 6 3]x 2 + 6x + 8 4]x 2 - 6x + 8 The answer =( x – 3 ) ( x – 2 ) =( x + 3 ) ( x + 2 ) =( x+ 2 ) ( x + 4 ) =( x – 2 ) ( x – 4 )

Factorization of a trinomial with the form ax 2 ± bx + c a ≠ 1 Factorize: 2 x 2 + 5x + 3 Step 1: open two brackets and put ( half power of first term ) x in each (x ) (x ) Step 2: the sign in the brackets are the same sign of the middle term ( x + ) ( x + ) Step 3: search about two numbers the product of them (ac) is 6 and their sum (b) is 5 2 and 3 2 x 2 + 5x + 3 = ( x + 2 ) ( x + 3 ) ( x + 2/2 ) ( x + 3/2 ) ( simplify if is it possible ) ( x + 1 ) ( 2X + 3 )

1)Try by your self (on 12 min. ) Factorize the following four question :- 1] 4x x 2 + 6x 3] 3x x 2 + 8x 2] 9x 2 – 6x +1 4] 5 + 7b + 2b 2 The answer x(4 x - 3 ) ( x - 2 ) ( 3x - 1 ) ( 3x - 1 ) X( 3x + 2 ) ( x + 4 ) 2b 2 + 7b + 5 = ( b + 1 ) ( 2b + 5 )

H.W. page 4 no. ( 1,2,4) page 6 second no. ( 1,4,5)

case one if the sign of the last term is negative is (-ve): Ex (1): Factorize: X 2 - 5X -6 Step 1: open two brackets and put ( half power of first term ) x in each (x ) (x ) solution Step 2: the sign in the brackets are always different ( x - ) ( x + ) Step 3: search about two numbers the product of them (ac) is 6 and their diferrence (b) is 5 6 and 1 X 2 - 5X -6 = (x-6) (x+1) Step 4: put the greatest no. with the same singe of middle term. ( -6) (+1) Factorization of a trinomial with the form ax 2 ± bx – c a = 1

1)Try by your self (on 12 min. ) Factorize the following four question :- 3] x 2 -x -6 1]x 2 - 4x -12 2] x 2 + 4x ] a 2 -a -30 The answer ( x – 6 ) ( x + 2 ) ( x + 6 ) ( x – 2 ) ( x + 2 ) ( x – 3 ) ( a – 6 ) ( a + 5 )

Factorization of a trinomial with the form ax 2 ± bx – c a ≠ 1 Factorize: 2 x 2 + 5x – 3 Step 1: open two brackets and put ( half power of first term ) x in each (x ) (x ) Step 2: the sign in the brackets are always different ( x - ) ( x + ) Step 3: search about two numbers the product of them (ac) is 6 and their difference (b) is 5 1 and 6 2 x 2 + 5x + 3 = ( x + 6 ) ( x - 1 ) ( x + 6/2 ) ( x - 1/2 ) ( simplify if is it possible ) ( x + 3 ) ( 2X - 1 ) Step 4: put the greatest no. with the same singe of middle term. ( +6) (-1)

3] 6x 2 – x –2 2] 7x 2 – 4x – 3 1] 2x 2 + 3x – 2 1)Try by your self (on 9 min. ) Factorize the following three question :- The answer ( x + 2 ) ( 2x - 1 ) ( x - 1 ) ( 7x + 3 ) ( 2x + 1 ) ( 3x – 2 )

H.W. page 4 no. ( 3,6,12) page 6 second no. ( 7,8,10)

Factorization of a perfect square The trinomial expression is said to be perfect square if : Which of these expressions is a perfect square? a)x 2 – 6x – 9 b) x 2 – 6x + 9 c) a (a – 8) + 16 d) 3x 2 – 18x + 27 YesNo Yes Yes Yes No No No

Factorize: a) 4x 2 – 12x + 9 (2x – 3) 2 b) 18x x x 2x (9x x + 16) ( take H.C.F.) 2x (3x + 4) 2

1)Try by your self (on 9 min. ) Factorize the following three question :- 1]1+ 14x +49x 2 3]50x xy +2y 2 = 2}4a a + 25 The answer (2a – 5) 2 (1 + 7x) 2 2(5x – y) 2

H.W. page 10 no. 4( a,b,c)

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