موضوع: 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
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2] a positive whole number , if its square is added to its three times then the result will be 10 . what is the number ?
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3] Find the number which if its added to its square , the result will be 24 .
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4] two consecutive positive odd numbers , the sum of their squares is 650 what are the two numbers ?
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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 ?
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6] the length of a rectangle is 2 cm more than its width . if its area is 360cm2 find its perimeter .
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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 .
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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 .

1- Factorize by taking H.C.F
A] 6 x2y2 – 9 x3 y by identifying H.C.F
H.C.F is 3x2y
3x2y ( 2 y – 3 x )
B] 20a3b2 + 4a4b3-24ab3 ……………………………………
…………………………………………………………………………………………………

2- Factorize the expression in the form
a X 2 + b X + c where a = 1
when you are going to factorize any expression follow : ـ
1- arrange the expression descending according to the
power of X
2- is there a common factor between the terms or no
3- find two numbers their product is = c
And their sum = b
Find :
Two numbers their product is 20 and their sum is 9
Two numbers their product is -24 and their sum is 5
Two numbers their product is 12 and their sum is -8
Two numbers their product is -15 and their sum is -14
Example :
A] Factorize x2 - 5 x + 6
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B] Factorize x2 +11 x +10
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C] Factorize x2 - 7 x + 10
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D] Factorize y2 – 50 y - 51
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E] Factorize - x2 + 2 x + 63
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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
……………………… ……………………..

Remark when you going to factorize any expression
1] arrange the expression descendingly according to the power of x
2] Factorize firstly by taking H.C.F if exist
And if the expression in the form of ax2 b x + c when a 1
Example (1)
Factorize each of the following 3 x2 + 7 x - 6
The first term 3 x2 = 3x X x
The last term - 6 can be factorize to
-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






In first try
3x X -6 + x X 1 = -17x the middle term
In the second try
3x X 6 + x X -1 = 17 x the middle term
In the third try
3 x X - 3 + x X 2 = -7x the middle term
In the last try
3 x X 3 + x X -2 = 7x = the middle term
3 x2 + 7x – 6 = ( 3x – 2 ) ( x + 3 ) take care and follow the mister

Exercises on factorization a trinomial with the form
ax2 b x + c when a 1
1] factorize each of the following expressions
A] 2a2 + 5a + 2
B] 2x2 -5x + 2
C] 3x2 +5x – 12
D] 4y2 + 5y -21
2] factorize each of the following expressions
A] 6a2 + 5ab + b2
B] 6a2 -19ab-7b2
C] 25 m-10 + 15 m2
D] 8x3-27x2 -20x
E] 6x3 -13 x2y +6xy2
F] 4x ( 3x +7y ) -5y2