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**berber** · الأربعاء 20 أغسطس - 0:40:02

[ltr] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image002$ [/ltr]

[ltr]1- Factorize by taking H.C.F [/ltr]
[ltr] A] 6 x²y² – 9 x³ y                          by identifying H.C.F [/ltr]
[ltr]                                                               H.C.F is 3x²y [/ltr]
[ltr]       3x²y ( 2 y – 3 x )   [/ltr]
[ltr]B] 20a³b² + 4a⁴b³-24ab³              ……………………………………[/ltr]
[ltr] …………………………………………………………………………………………………[/ltr]
[ltr] [/ltr]
[ltr]2- Factorize the expression   in the form [/ltr]
[ltr]a X ² + b X + c    where a = 1 [/ltr]
[ltr]when you are going to factorize any expression follow : ـ [/ltr]
[ltr]1-     arrange the expression descending according to the [/ltr]
[ltr]                    power of X [/ltr]
[ltr]2- is there a common factor between the terms or no [/ltr]
[ltr]3- find two numbers their product is = c [/ltr]
[ltr]         And their sum = b [/ltr]
[ltr] Find : [/ltr]
[ltr]   Two numbers their product is 20 and their sum is 9 [/ltr]
[ltr]   Two numbers their product is -24 and their sum is 5 [/ltr]
[ltr]   Two numbers their product is 12 and their sum is -8 [/ltr]
[ltr]   Two numbers their product is -15 and their sum is -14 [/ltr]
[ltr]Example :[/ltr]
[ltr]                A]    Factorize   x² - 5 x + 6 [/ltr]
[ltr]                       ……………………………………………………………………………[/ltr]
[ltr]               B]    Factorize   x² +11 x +10 [/ltr]
[ltr]                       ……………………………………………………………………………[/ltr]
[ltr]                C]    Factorize   x² - 7 x + 10 [/ltr]
[ltr]                       ……………………………………………………………………………[/ltr]
[ltr]                D]    Factorize y² – 50 y - 51 [/ltr]
[ltr]                        ……………………………………………………………………………[/ltr]
[ltr]                E]    Factorize - x² + 2 x + 63 [/ltr]
[ltr]……………………………………………………………………………[/ltr]
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[ltr] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image004$ [/ltr]
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[ltr]1] Factorize each of the following [/ltr]
[ltr]       a) x² + 8 x + 15   …………………………………………………………………………………………[/ltr]
[ltr]       b) a² – 9 a + 18   …………………………………………………………………………………………[/ltr]
[ltr]       c) a²- a - 30      …………………………………………………………………………………………[/ltr]
[ltr]       d) a² + 30 a + 81 …………………………………………………………………………………………[/ltr]
[ltr]       e) y² + 7 y + 1 2 …………………………………………………………………………………………[/ltr]
[ltr]2] Factorize each of the following [/ltr]
[ltr]       1) x² + 5 xy + 6y² ………………………………………………………………………[/ltr]
[ltr]       2) y² + 16 yx – 36 y²………………………………………………………………………[/ltr]
[ltr]       3) a² + 22 ab – 48 b²………………………………………………………………………[/ltr]
[ltr]       4) n² + 4 nm – 45 m² ………………………………………………………………………[/ltr]
[ltr]       5) x² – 7 xy – 18 y²………………………………………………………………………[/ltr]
[ltr]       6) x² – 5xy - 24 y²   ……………………………………………………………………………[/ltr]
[ltr]3] Factorize each of the following [/ltr]
[ltr]       1) 15 a + a² – 34      …………………………………………………………………………………[/ltr]
[ltr]       2) -10 + x² + 3x       ……………………………………………………………………………[/ltr]
[ltr]       3) 22 x - 7 5 + x²     ……………………………………………………………………………………[/ltr]
[ltr]       4) x² + 21 – 10 x       ………………………………………………………………………………[/ltr]
[ltr]4]   Factorize each of the following[/ltr]
[ltr]         1) 2 a² + 28 a +96 ………………………………………………………………………………………… 2) 2x² + 60 x + 162 …………………………………………………………………………[/ltr]
[ltr]       3) x³ – 23 x² + 60 x ………………………………………………………………………………[/ltr]
[ltr]       4) 3x² – 42 – 15 x    …………………………………………………………………………………[/ltr]
[ltr]5] Factorize each of the following[/ltr]
[ltr]       1) x⁴ + 9 x² – 36        …………………………………………………………………………………………[/ltr]
[ltr]       2) x⁶ + 8x³y³ – 180 y⁶ …………………………………………………………………………………………[/ltr]
[ltr]       3) a¹⁰ – 9a⁵b³ – 220 b⁶ ………………………………………………………………………………………[/ltr]
[ltr]6] Find the positive integer of a which makes each of the following [/ltr]
[ltr]    Expressions factorizable [/ltr]
[ltr]       1) x² + 5x + a               2) x² + a x - 2 [/ltr]
[ltr]           ……………………….                    ………………………..[/ltr]
[ltr]       3) x² + ax + 2                 4) x² – a x + 12 [/ltr]
[ltr]          ………………………                     ……………………..[/ltr]
[ltr]                                                              [/ltr]

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[ltr] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image008$ [/ltr]

[ltr] [/ltr]

[ltr]Remark when you going to factorize any expression [/ltr]
[ltr]    1] arrange the expression descendingly according to the power of x [/ltr]
[ltr]    2] Factorize firstly by taking H.C.F if exist [/ltr]
[ltr]And if the expression in the form of ax² $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image009$ b x + c when a $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image010$ $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image009$ 1 [/ltr]
[ltr] Example (1) [/ltr]
[ltr]             Factorize each of the following 3 x² + 7 x - 6[/ltr]
[ltr]                    The first term 3 x² = 3x   X x [/ltr]
[ltr]                    The last term - 6 can be factorize to [/ltr]
[ltr]-6 x 1 or -1 x 6 or 2 x -3 or -2 x 3 then we try by using the following arrows to get the correct factorization[/ltr]


	$احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image012$

		$احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image013$

[ltr] [/ltr]
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[ltr]In first try [/ltr]
[ltr]3x X -6   + x X 1 = -17x $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image010$   the middle term [/ltr]
[ltr]In the second try [/ltr]
[ltr] 3x X 6 + x   X -1 = 17 x $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image010$   the middle term [/ltr]
[ltr]In the third try [/ltr]
[ltr] 3 x   X - 3 + x X 2 = -7x $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image010$ the middle term [/ltr]
[ltr]In the last try [/ltr]
[ltr] 3 x X 3 + x X -2 = 7x = the middle term [/ltr]
[ltr] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image015$ 3 x2 + 7x – 6 = ( 3x – 2 ) ( x + 3 ) take care and follow the mister [/ltr]
[ltr] [/ltr]
[ltr]Exercises on factorization a trinomial with the form [/ltr]
[ltr]      ax² $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image009$ b x + c when a $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image010$ $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image009$ 1 [/ltr]
[ltr]1] factorize each of the following expressions [/ltr]
[ltr]       A] 2a² + 5a + 2 [/ltr]
[ltr]       B] 2x² -5x + 2 [/ltr]
[ltr]       C] 3x² +5x – 12 [/ltr]
[ltr]       D] 4y² + 5y -21 [/ltr]
[ltr]2] factorize each of the following expressions[/ltr]
[ltr]       A] 6a² + 5ab + b² [/ltr]
[ltr]       B] 6a² -19ab-7b² [/ltr]
[ltr]       C] 25 m-10 + 15 m² [/ltr]
[ltr]       D] 8x³-27x² -20x[/ltr]
[ltr]       E] 6x³ -13 x²y +6xy²[/ltr]
[ltr]F] 4x ( 3x +7y ) -5y²[/ltr]
[ltr]Exercises from the school book [/ltr]
[ltr] [/ltr]
[ltr]1] Complete the missing terms[/ltr]
[ltr] [/ltr]
[ltr]    A] 2 x²+7 x-6 = ( 3x - …… ) ( …… + …… ) [/ltr]
[ltr]    B] 2 x² + x – 6 = ( …… + …… ) ( x - …… ) [/ltr]
[ltr]    C] 5 x²-2 x-7 = ( 5x - …… ) ( x + …… ) [/ltr]
[ltr]    D] 6 x²-11 x -10 = ( 2x - ……. ) ( …… + 2 )[/ltr]
[ltr]    E]2 x² + 10 x + 8 = ( …… + 4 ) ( x + …… ) [/ltr]
[ltr] [/ltr]
[ltr]2] Factorize each of the following expressions[/ltr]
[ltr]                                                     [/ltr]
[ltr]    A] 2 x² + 3x + 1                       B] 5z²-7z +2 [/ltr]
[ltr]    C] 5x²- 3x -3                          D] 8a²+ 2a -3[/ltr]
[ltr]    E] 6x² - 47xy - 63y2            F] 21x²y²+ 6x²y³- 15x²y⁴[/ltr]
[ltr]    G] 2y²+ 5y +3                         H] 6x²-11x + 3[/ltr]
[ltr]    J] 5 a²-18 a +1 6                            k] 2 m²- 9 m -5 [/ltr]
[ltr] [/ltr]
[ltr]3] a rectangle its area ( 2 x2 + 19 x + 35 ) cm2. Find its dimensions in terms of x then find its perimeter when x = 3 [/ltr]
[ltr] [/ltr]

[ltr]Factorization of a perfect square trinomial [/ltr]

[ltr] [/ltr]

[ltr]Each of the following expressions [/ltr]
[ltr] 4x² - 12 x +9           ,   25 y² + 70 xy + 49 x² [/ltr]
[ltr]          And L⁴ – 10 L²m + 25 m² are called perfect square [/ltr]
[ltr] [/ltr]
[ltr]And we note that any   expressions has :[/ltr]
[ltr]1-          each of the first term and the third term are perfect [/ltr]
[ltr]         square .[/ltr]
[ltr]2- the middle term = $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image017$ 2 $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image019$   x $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image021$ [/ltr]
[ltr] If the expression has the previous conditions is called [/ltr]

[ltr]perfect square.[/ltr]

[ltr] [/ltr]
[ltr]And the factorization of this expression in the form [/ltr]
[ltr] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image022$ ( $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image024$    $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image017$    $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image027$   ) ²[/ltr]
[ltr]            [/ltr]
[ltr]          The sign of the middle term [/ltr]


	$احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image032$

[ltr] [/ltr]
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[ltr]Firstly as any expression we should [/ltr]
[ltr]1)           take the H.C.F of the expression if exist [/ltr]
[ltr]2)         arrange the terms of the expression desceningly [/ltr]
[ltr]            according to the power of any symbol . [/ltr]
[ltr]1] Show which of the following expression is a perfect square [/ltr]
[ltr] A] a²   + 9                                       B] a² – ab + b² [/ltr]
[ltr] C] L² – 8Lm + 16 m²                        D] 4c⁴– 12 c²d - 9 d²[/ltr]
[ltr] E] 1 – a + a²                                    F] 4 + 36 a² + 81 a⁶[/ltr]
[ltr] [/ltr]
[ltr]2] Complete the missing term in each of the following [/ltr]
[ltr]               trinomial to be a perfect square[/ltr]
[ltr] A] 4a² + ………… + 25                             B] 9a² - ……… + 36 [/ltr]
[ltr] C] 4a² + ……… + 36 b2                            D] 36 x⁴ - ……… + 9y⁴[/ltr]
[ltr] E] a² – 6 a + ………                                   F] 4x²+28x + …………[/ltr]
[ltr] G] 4b² – 4a b + …………                              H] 25 m² – 10 mn + ….[/ltr]
[ltr] J] ………. – 36 y² + 9                              K] …… - 24 ab + 16 b²[/ltr]
[ltr]L] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image034$ x² + ……… + $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image036$ y²                             M] ……… -18y² + 81[/ltr]
[ltr]3] Choose the correct answer :[/ltr]
[ltr] A] if the expression x² + 14 x + B is a perfect square then [/ltr]
[ltr]         B= ………                    ( 2 , 7   , 14 , 49 ) [/ltr]
[ltr] B] if ( x + y ) ² = 64 , xy = 15 then x² + y² = ……..[/ltr]
[ltr]                                         ( 8 , 34 , - 34 , 49 ) [/ltr]
[ltr]C] if a² + b² = 11 , ab = 15 then a²-b² = …………[/ltr]
[ltr]                                       ( 6a , $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image009$ 1 , 1    , - 1 )[/ltr]
[ltr]D] (99)² + 2( 99 ) + 1 = ………….( 100 ,10 000 , 410 , (98)² )[/ltr]
[ltr]E] if a² +2ab +b² = 25 then a + b = ……( 5 , -5   , $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image009$ 5, 12.5 )   [/ltr]
[ltr]F] if x² + Kx + 25 is a perfect square then K = ……[/ltr]
[ltr]                                               ( 5 , 10 , $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image009$ 10 , $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image009$ 5 ) [/ltr]
[ltr]4] Factorize each of the following [/ltr]
[ltr] A] x² + 2x y + y²                            B] a²-2 ab + b² [/ltr]
[ltr] C] 49 a² – 56 a + 16                       D] 4x² – 20 x + 25 [/ltr]
[ltr] E] 1 – 10 a² + 25 a⁴                         F] -4m²n² + m⁴ + 4n⁴[/ltr]
[ltr]5] Factorize each of the following[/ltr]
[ltr] A] 2a² + 28 a + 98                          B] 3a² – 6ab +3b² [/ltr]
[ltr] C] 8x²-4x⁴ -4                         D] 12x² + 36xy +27y²[/ltr]
[ltr] E] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image039$ a² + $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image041$   a + $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image034$                       F] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image044$ x² - $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image041$ x + $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image047$ [/ltr]
[ltr] G] 4x² – 7y ( 4x-7y )                       H] 9a² + 5b ( 5b – 6a ) [/ltr]
[ltr]6] using factorization to find the value of [/ltr]
[ltr]                                  A] (20.7) ² – 1.4 x 20.7 + ( 0. 7 )²        [/ltr]
[ltr]                                 B] ( 99 ) ² –2(99) (98) + ( 98 )² [/ltr]

[ltr] [/ltr]

[ltr]Factorization of the difference of two squares [/ltr]

[ltr]We know that [/ltr]
[ltr] (x + 3 ) ( x – 3 )        = X² – 9 [/ltr]
[ltr] ( a+ b ) ( a-b)          = a² – b²[/ltr]
[ltr]Each of X² – 9   and a² – b² are called difference between two squares and so on 25 x² – 16 y² and ( 75 ) ² – ( 25)²[/ltr]
[ltr] And the difference between two squares [/ltr]
[ltr]                        = their sum x the difference between them[/ltr]

[ltr]Examples[/ltr]

[ltr]1] Factorize each of the following [/ltr]
[ltr]          A] 49x² - 25   = ( 7x + 5) ( 7x – 5 ) [/ltr]
[ltr]          B ] (2y-3) ² - 1 = [ ( 2y -3 ) – 1 ] [ ( 2y -3 ) + 1 ] [/ltr]
[ltr]          C] 27 m³ – 48 mn⁶           H.C.f 3m [/ltr]
[ltr]                  = 3m ( 9 m² - 16 n⁶ ) [/ltr]
[ltr]                  = 3m ( 3m – 4n³ ) (3m + 4 n³) [/ltr]
[ltr]          D] 81 x⁴ – 16 y⁴       = ( 9x² + 4y² ) (9x²- 4y²) [/ltr]
[ltr]                                         = (9x² + 4y²) ( 3x-2y ) (3x+2y)[/ltr]
[ltr]2] Factorize each of the following[/ltr]
[ltr]          A] x² - 4                        B] 16x² – 9 [/ltr]
[ltr]          C] 49x²-64y²                          D] 169x² -144y²[/ltr]
[ltr]          E] x² – 1                                  G] x⁴ – a ⁶[/ltr]
[ltr]          H] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image049$ y²- $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image051$                             j] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image053$    - $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image055$ [/ltr]
[ltr]3] Factorize each of the following[/ltr]
[ltr]          A] 2x² – 50                             B] 3y² -27 [/ltr]
[ltr]          C] x³-x                                    D] x³y – xy⁵ [/ltr]
[ltr]          E] x²(x+2y ) – y² ( x+2y )        F] 4b² (2a-b) -25a²(2a-b)[/ltr]
[ltr]4] using factorization to find the value of each of the following[/ltr]
[ltr]        A] (33)² – ( 23)²                     B] (77)² – ( 23)²[/ltr]
[ltr]        C] (8.27)² – ( 1.23)²                D] 31 x 29 [/ltr]
[ltr]5] if x² –y² = 20 , x + y = 10 Find x-y [/ltr]
[ltr]6] if L – M = 9 , L + M = 15 then L² – M² = ………................................[/ltr]

[ltr] [/ltr]

[ltr]Factorization of sum and difference between two cubes [/ltr]

[ltr] a ³ + b³ is called sum of two cubes and [/ltr]
[ltr] x³ – y³ is called difference between two cubes[/ltr]
[ltr]Study the following table[/ltr]
[ltr] [/ltr]

[ltr]X[/ltr]	[ltr]1[/ltr]	[ltr]2[/ltr]	[ltr]3[/ltr]	[ltr]4[/ltr]	[ltr]5[/ltr]	[ltr]6[/ltr]	[ltr]7[/ltr]	[ltr]8[/ltr]	[ltr]9[/ltr]	[ltr]10[/ltr]
[ltr]X²[/ltr]	[ltr]1[/ltr]	[ltr]4[/ltr]	[ltr]9[/ltr]	[ltr]16[/ltr]	[ltr]25[/ltr]	[ltr]36[/ltr]	[ltr]49[/ltr]	[ltr]64[/ltr]	[ltr]81[/ltr]	[ltr]100[/ltr]
[ltr]X³[/ltr]	[ltr]1[/ltr]	[ltr]8[/ltr]	[ltr]27[/ltr]	[ltr]64[/ltr]	[ltr]125[/ltr]	[ltr]216[/ltr]	[ltr]343[/ltr]	[ltr]512[/ltr]	[ltr]729[/ltr]	[ltr]1000[/ltr]

[ltr] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image056$ [/ltr]
[ltr] [/ltr]
[ltr] [/ltr]
[ltr] [/ltr]
[ltr] [/ltr]
[ltr] [/ltr]

[ltr]Examples [/ltr]

[ltr]1 ] Factorize each of the following [/ltr]
[ltr]A] 8 x³ + 27   = ( 2x + 3 ) ( 4x² -6x + 9 ) [/ltr]
[ltr]B] 125 a³ – b⁶ = ( 5a – b² ) ( 25a² + 5ab²+ b⁴ ) [/ltr]
[ltr]C] x³ + 343 y³ = ( x + 7y ) ( x² – 7xy + 49y² ) [/ltr]
[ltr]D] 40 a³ + 135 b³ …………………… H.C.F is 5 [/ltr]
[ltr]        5( 8a³ + 27b³) = 5( 2a + 3b )( 4a² -6ab + 9b²) [/ltr]
[ltr]E] x⁶ – 64y⁶   For excellent pupils !!!!!!!!!![/ltr]
[ltr] [/ltr]
[ltr]2] Factorize each of the following [/ltr]
[ltr]        A] x³ + 1                                         B] x³ - 8 [/ltr]
[ltr]C] 125 + a³                                      D] 343 – 2 7 x³ [/ltr]
[ltr]E] L³ - $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image058$                                       F] 8a³ + 0. 0 0 1 [/ltr]
[ltr]G] 0 . 0 27 x³ + $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image060$ y³                    H] y⁶ – 1 [/ltr]
[ltr]I] x⁶ + y ⁶                                     J] a³ – 0 . 008 b³ [/ltr]
[ltr]K] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image062$                                         L] 27 x³y³ - 64[/ltr]

[ltr]3] Factorize each of the following [/ltr]
[ltr]A] 2x³ + 16                                     B] 3x³ – 81 [/ltr]
[ltr]C] L⁴ – 27 L                                    D] 16 x³ + 250 y³ [/ltr]
[ltr]E] 216 a³ – 27 b³                            F] 54 x⁴ y² – 16 xy⁵[/ltr]
[ltr]4] Complete [/ltr]
[ltr]1) $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image064$   = …………                     2) $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image066$     = …………[/ltr]
[ltr]3) 27 m³ = ( ……… ) ³              4) 343 x³ y⁶= ( …………… ) ³[/ltr]
[ltr]5) x³– 1 = ( x -1 ) ( …………………) [/ltr]
[ltr]6 ) 8 a³ + 125 = ( …………… ) ( 4a2 - 10 a + ……… ) [/ltr]
[ltr]5] if x³ – y ³ = 28 , x - y = 2 Find [/ltr]
[ltr]        The value of x² + xy + y² [/ltr]
[ltr]6] Choose the correct answer [/ltr]
[ltr]1) if x – y = 2 , x² + xy + y² = 4 then x³ – y³ = ………….[/ltr]
[ltr]    A ) 2                  b) 4                 c) 6                 d) 8 [/ltr]
[ltr]2) if a³ + b³ = 35 , a² –ab + b² = 7 then a + b = …………[/ltr]
[ltr]A ) 5                  b) 28               c) 42               d) 245[/ltr]
[ltr]3) if y³ – a = ( y – 2 ) ( y² + 2y + 4 ) then a = …………[/ltr]
[ltr]A ) 2                  b) 4                 c) -8                        d) 8 [/ltr]
[ltr]4) if x³ – 8 = ( x + a ) ( x² + 2x + 4 ) then a = ………….[/ltr]
[ltr]A ) -4                         b) 4                 c) 26                       d) -2[/ltr]
[ltr]5) ( x³– 64 ) ÷ ( x- 4 ) = …………………… where x $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image010$ 4 [/ltr]
[ltr]A ) x² + 16    b) x² – 16 c) x²- 4x +16       d) x² + 4x + 16[/ltr]
[ltr] [/ltr]

[ltr] [/ltr]

[ltr]Factorization by grouping[/ltr]

[ltr]In this case the expression consisting of four terms is factorized by group each two terms which have a common factor.[/ltr]

[ltr]Solved example [/ltr]
[ltr]1) Factorize   2a³ + 5a² + 8a + 20 [/ltr]
[ltr]                        = (2a³+ 5a² ) + ( 8a + 20 ) [/ltr]
[ltr]                        = a² ( 2a + 5 ) + 4 ( 2a + 5 ) [/ltr]
[ltr]                        = ( 2a + 5 ) ( a² + 4 ) [/ltr]
[ltr]2) Factorize x³ – 3 x² – 5x + 15 [/ltr]
[ltr]                        = ( x³ – 3 x² ) + ( 15 – 5x ) [/ltr]
[ltr]                        = x2 ( x – 3 ) + 5 ( 3 – x ) [/ltr]
[ltr]                        = x2 ( x – 3 ) – 5 ( x – 3 ) [/ltr]
[ltr]                        = ( x – 3 ) ( x2 – 5 ) [/ltr]
[ltr]3 ) Factorize 16 x² – a² + 6ab – 9 b ²  [/ltr]

[ltr] We note that there is no common factor between the [/ltr]

[ltr] first term and the other terms so we can group as [/ltr]

[ltr] $احلى ملزمه فى math alg factorization C:\Users\berber\AppData\Local\Temp\msohtmlclip1\01\clip_image068$ 16 x² + (– a² + 6ab – 9 b ²) ( 16 x²-(- a² - 6ab +9 b ²) [/ltr]

[ltr] 16 x² – ( a – 3b ) ² difference between two squares[/ltr]

[ltr] ( 4x – ( a -3 b ) ) ( 4x + ( a -3 b ))[/ltr]

[ltr] ( 4x –a + 3b ) ( 4x + a -3 b ) [/ltr]

[ltr]Exercise[/ltr]

[ltr]1] Factorize each of the following completely [/ltr]
[ltr]        A] ab + ac + b + c                    B] ax – cy – cx + a y [/ltr]
[ltr]        C] 2ab – 10 a + 3b – 15            D] ab + 5b + 7a + 35 [/ltr]
[ltr]        E] 5 L - 10 m – al + 2 am        F] x³ – 3x² + 6 x - 18 [/ltr]
[ltr]2] Factorize each of the following completely [/ltr]
[ltr]        A] y³ + 2 y ² – y – 2                 B] x5 – x3 – x2 + 1 [/ltr]
[ltr]        C] 30 x2y² – 3 x y + 6x²y – 15 xy²[/ltr]
[ltr]        D] a³ + b³ – a – b                     E] abx² + b x – a x – 1 [/ltr]
[ltr]        F] 8mn – 2 m² + 12 n L – 3mL [/ltr]

الموضوعالأصلي : احلى ملزمه فى math alg factorization // المصدر : ممنتديات جواهر ستار التعليمية //الكاتب: berber

**berber** · الأربعاء 20 أغسطس - 0:40:57

احلى ملزمه فى math alg factorization

Summery of the factorization

The expression

1- Taking H.C.F ab + ac = a ( b + c )
6x2y + 10 xy2 = 2xy ( 3x + 5y )
2a ( x + y ) – b ( x + y ) = ( x + y ) ( 2a – b )

2- Difference between two squares a2 - b2 = ( a+b) ( a-b)
X2 – 9 = ( x+3) ( x – 3 )
2x3 – 72x = 2x ( x2-36)
= 2x ( x + 6 ) ( x – 6 )

3- Sum of two cubes a3+ b3 = ( a+b ) ( a-ab+b2)
Difference between two cubes a3-b3 = ( a-b) ( a + ab –b2)
X3 + 8 = ( x + 2 ) (x2 – 2x + 4 )
X6 – 64y6 = ( x3 + 8y3) ( x3 – 8y3)
= ( x + 2 y ) ( x2 – 2xy + 4 y2)
X ( x – 2y ) ( x2 + 2xy +4y2 )

4- The perfect square trinomial a2 + 2ab + b2
X2 + 10 x +25 = ( x + 5 ) 2

The trinomial in the form of x2 + bx + c
X2 + 7x + 12 = ( x +3 ) ( x + 4 )
18 x – 15x2 + 3 x3 = 3x3 -15x2+ 18x order
= 3x ( x2 – 5x + 6 ) H.C.F
= 3x ( x -2 ) ( x -3 )

5- Factorizing by grouping
Ax + ay + b x +by = a ( x + y ) + b ( x + y )
= ( x + y ) ( a + b )
X2 – 2 x y + y2 – 9 = x 2 – 2 x y + y2 ) - 9
= (x – y )2 - 9
= ( x – y + 3 ) ( x – y – 3 )

Solving an equation of the second degree
in one variable algebraically

Find the solution set of x ( x – 1 ) = 0
Either x = 0
Or x -1 = 0
x = 1
S.S = { 0 , 1 }
Find S.S of x( x+1 ) ( x – 1 ) = 0
Either x = 0 or x + 1 = 0 or x – 1 = 0
x = -1 x = 1
the S.S = { 0 , 1 , - 1 }

Find the S.S of ( x2 – 4 ) ( x2 +1)
Either x2 – 4 = 0 or x2 + 1 = 0
x = 2 x2 = -1 has no root
the S.S = { 2 , - 2}
Find in Q the S.S of each of the following equations
1] x2 – 9 = 0 First we should factorize the L.H.S
( x + 3 ) ( x – 3 ) = 0 then either x + 3 = 0 or x-3 = 0
x = -3 x = 3
The S.S = { 3 , - 3 }
2] x2 + 8 x + 12 =0
(x + 6 ) ( x + 2 ) = 0 either x + 6 = 0 or x + 2 = 0
x = -6 x = -2
The S.S = { - 6 , - 2 }

Exercises
1] Find in Q the S.S of each of the following equations
1) x2 – 4x -21 = 0
2) x2 + 4 x + 4 = 0
3) 2x2 + 2x – 35 = 0
4) 9x2 – 6 x + 1 = 0
5) x2 - 7x – 30 = 0
2] Find in Q the S.S of each of the following equations
1) x2 = x 2) 3x2 = 7x
3) 4x2 = 49 4) x2 – x = 6
5) 4x3 = 9 x 6) 6x2 –x = 22
3] Find in Q the S.S of each of the following equations
1) x ( x – 5 ) + 6 = 0 2) x ( x + 3 ) -18 =0
3) ( x + 8 ) ( x – 3 ) = 3 x 4) ( x + 3 ) 2 – 49 = 0
5) 5 ( x2 + 3 ) = 6 0 6) 4( x+4)2 = 49

Word problems of solving quadratic equations
Firstly you should understand the following table
IF Then
The number = x • Its half = x or
• Its third = x or
• Its double ( twice ) = 2 x
• Its three times = 3 x
• Its square = x2
• Double its square = 2x2
• Square its double = (2x)2 = 4x2
• Its additive inverse = - x
• Its multiplicative inverse =

Three consecutive numbers • The 1st no is x
• The 2nd is x + 1 and the 3rd is x +2
Three consecutive even ( odd ) numbers • The 1st no is x
• The 2nd is x + 12 and the 3rd is x +4
Two numbers the ratio between them is 2 : 3 • The 1st no is 2 x and
• the 2nd no is 3x
Square its side length is x cm • Its perimeter = 4 x cm
• Its area = x2 cm2

Rectangle its length exceeds than its width by 5 cm

• The width = x and the length = x +5
• Its perimeter = ( x + x + 5 ) x 2
• Its area = x ( x + 5 )
A man his age now is x years • His age after 4years is x + 4
• His age 4 years ago = x – 4
Two numbers one of them more than the other by 5 • The 1st is x
• the second is x + 5 or x – 5 think
why ?
Two numbers one of them greater than twice the second by 5 • the 1st is x
• The 2nd is 2 x + 5

Solved problems
1] two real numbers one of them increase than the other by 4
If the product of the two numbers is - 45 .
what is the two numbers ?
let the smallest number is x then the other is x + 4
x ( x + 4 ) = 45 equation
x2 + 4 x - 45 = 0 ( x + 9 ) ( x – 5 ) = 0 then
either x + 9 = 0 x = - 9
or x – 5 = 0 x = 5
2] if Hatem’s age now exceeds Hanan’s age by 4 years , and the sum of the squares of their ages now is 26 years .
Calculate the age of each .
Let Hanan ‘sage is x then Hatem’s age is x + 4
then her square is x2 then his square ( x + 4 ) 2
x2 + ( x+4)2=26 x2 + x2+8x+16=26
2x2 + 8x +16-26 = 0 2x2 + 8x -10 = 0
2( x2 + 4x -5 ) =0
x2 +4x -5 = 0
(x+5 ) ( x-1) = 0
3] apiece of land in a rectangular shape its length increase
Its width by 5 m . if its area is 500m2
Find its dimensions .
Solution
Let the width is x then its length is x + 5
Area of the rectangle = L x W
500 = ( x + 5 ) x
500 = x2 +5x
x2 + 5x = 500 x2 + 5x -500 = 0
( x + 25 ) ( x – 20 ) = 0
Either x + 25 = 0 then x = - 25 refused
Or x - 20 = 0 then x = 20
The widths = 20 cm then the length = 20+5 =25 cm
4] a real number exceeds than its multiplicative inverse by
What is the number ?
Solution
Let the number is x then its multiplicative inverse is
x - = by multiply both sides by 6x why ? discuss
6x (x - ) = 6x ( )
6x2 – 6 = 5 x 6x2 – 5 x - 6 = 0
( 2x – 3 ) ( 3x + 2 )=0
Either 2x – 3 = 0 then 2x = 3 then x =
or 3x + 2 = 0 then 3x = -2 then x =
then the number = or

Exercises on word problems of
Solving quadratic equation
1] Find the rational number whose four times its square is 81
………………………………………………………………………………………………………
………………………………………………………………………………………………………
2] a positive whole number , if its square is added to its three times then the result will be 10 . what is the number ?
………………………………………………………………………………………………………
………………………………………………………………………………………………………
3] Find the number which if its added to its square , the result will be 24 .
………………………………………………………………………………………………………
………………………………………………………………………………………………………
4] two consecutive positive odd numbers , the sum of their squares is 650 what are the two numbers ?
………………………………………………………………………………………………………
………………………………………………………………………………………………………
5] Ali’s age is 4 years more than Omar’s age if the sum of the squares of their ages is 136 . what is the age of each ?
………………………………………………………………………………………………………
………………………………………………………………………………………………………
6] the length of a rectangle is 2 cm more than its width . if its area is 360cm2 find its perimeter .
………………………………………………………………………………………………………
………………………………………………………………………………………………………
7] ABC is a triangle in which m A =( x2 +61 ) ه ,
m B =(110 -11x) ه m C= ( 90 – 7x ) ه find the value of x then
calculate the measures of the angles of the triangle .
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………

General exercises from the school book
1] Factorize each of the following
A) x4 – 16 y4 B) 2x5 + 54 x2 C) a4 + 4b 4
D) x6 – 64y6 E)8x3 -125 F) 3x3+2x2+12x+8

2] Factorize each of the following
A)8 x2 -2xy – y2 B) L3m -27m4
C)625 a2 -81b2 D) 2(x +3y)3-250
E) (c – d ) + 2x ) c – d ) + x2 (c-d )
F) 7x2 -29xy+30y2

3] Find the value of C that makes the expression can be
factrizable . then factorize it . where C Z
A)x2 + C x -15 B) x2 -7x + C C) y2 –C y +29
D) a2 + a - C E) C x2 + x – 15 F) Cx2 -13x + 6

4] Factorize each of the following
A) 9x2 – 30 x + 25 B) 18ab4- 114b2c2a + 128ac2
C) x2 – 4xy + x – 2y + 4 y2 D) x2 – 2xy + y2 - 4H2

5] Find the S.S in R for each of the following equations
A) x 2 + x = 6 B) 3x2 + 2x = 85
C) (x – 1 )2 + x = 3 D) 2x3 = 7 x

6] three consecutive integer numbers their sum is the square of the middle number . find these numbers ?

7] In the opposite figure { c }
If m BCD = x2 , m ACD = 8x
Find the value of x

Unit test
School book
1] Choose the correct answer
A) the expression 4x2 + k + 25 y2 is a perfect square when
K = ……… ( 20 , 10 xy , 20 xy , 30 xy )
B) if x 2 - y2 = 16 , x + y = 8 then x – y = ………
( 2 , 1 , 128 , 6 4 )
C) if x + y = 3 , x 2 – xy + y2 = 5 then x 2 + y 2 = ……………
( 15 , 25 , 8 , 7 )
D) the expression 4 x2 + 12 x + a is a perfect square when a=
( 6 , 16 , 1 , 9 )
E) if ( 2a -5 ) ( 3a -2 ) = 6a2 + ka + 10 then k = …………
( 15 , 19 , -19 , 4 )
2] Complete
A] ( 4 a – 5 b ) ( ……… - 3 b ) = 8 a2 ………… + 15 b2
B) if x2 + y2 = 17 , xy = 7 then ( x-y) 2 = …………
C) if k x2 – 10 x + 1 is a perfect square then k = ………
D) if ( x + 1 ) is one of the factors of 5 x2 - 2x – 7 then
The other factor is …………………
E) x3 + 8 = ( x + 2 ) ( …………………… )
3] Factorize each of the following
A) ( x + 2 ) 3 – 4 x -8 B) a2+2ab +b2-c2
C) 2x2 – 5x + 3 D) x4 +4 L4
E) 8x3 – 343 y6
4] Find the S.S of each of the following equations
A) x2 – 3x -10 B) 3x2 + x = 14 C) ( 2x-1)2+(x-1)2=10
5] By using the factorization find the value of each
A) B) ( 8.175)2 - ( 1 .825 ) 2
C) (87)2 + 2 x 13 x 87 + (13)2
6] a right angled triangle the length of the sides of the right angle are 4x , x + 1 , if its area is 84 cm2 . find the length of the hypotenuse .

الموضوعالأصلي : احلى ملزمه فى math alg factorization // المصدر : ممنتديات جواهر ستار التعليمية //الكاتب: berber

**berber** · الأربعاء 20 أغسطس - 0:44:21

احلى ملزمه فى math alg factorization

1] Factorize each of the following
a) x2 + 8 x + 15 …………………………………………………………………………………………
b) a2 – 9 a + 18 …………………………………………………………………………………………
c) a2- a - 30 …………………………………………………………………………………………
d) a2 + 30 a + 81 …………………………………………………………………………………………
e) y2 + 7 y + 1 2 …………………………………………………………………………………………
2] Factorize each of the following
1) x2 + 5 xy + 6y2 ………………………………………………………………………
2) y2 + 16 yx – 36 y2 ………………………………………………………………………
3) a2 + 22 ab – 48 b2 ………………………………………………………………………
4) n2 + 4 nm – 45 m2 ………………………………………………………………………
5) x2 – 7 xy – 18 y2 ………………………………………………………………………
6) x2 – 5xy - 24 y2 ……………………………………………………………………………
3] Factorize each of the following
1) 15 a + a2 – 34 …………………………………………………………………………………
2) -10 + x2 + 3x ……………………………………………………………………………
3) 22 x - 7 5 + x2 ……………………………………………………………………………………
4) x2 + 21 – 10 x ………………………………………………………………………………
4] Factorize each of the following
1) 2 a2 + 28 a +96 ………………………………………………………………………………………… 2) 2x2 + 60 x + 162 …………………………………………………………………………
3) x3 – 23 x2 + 60 x ………………………………………………………………………………
4) 3x2 – 42 – 15 x …………………………………………………………………………………
5] Factorize each of the following
1) x4 + 9 x2 – 36 …………………………………………………………………………………………
2) x6 + 8x3y3 – 180 y6 …………………………………………………………………………………………
3) a10 – 9a5b3 – 220 b6 ………………………………………………………………………………………
6] Find the positive integer of a which makes each of the following
Expressions factorizable
1) x2 + 5x + a 2) x2 + a x - 2
………………………. ………………………..
3) x2 + ax + 2 4) x2 – a x + 12
……………………… ……………………..

الموضوعالأصلي : احلى ملزمه فى math alg factorization // المصدر : ممنتديات جواهر ستار التعليمية //الكاتب: berber

**wassim25** · الثلاثاء 2 سبتمبر - 20:10:33

احلى ملزمه فى math alg factorization

شكرا يبا ياغالي علي الموضوع

الموضوعالأصلي : احلى ملزمه فى math alg factorization // المصدر : ممنتديات جواهر ستار التعليمية //الكاتب: wassim25

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