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Chapter:5
Factoring
Chapter :5
Topics
Formula and quadratic Equations using Factoring
Factoring Monomials with a 1
Factoring Monomials with a = 1
Factoring by Grouping
Factoring a Monomial from a Polynomial
Chapter :5
Hot TipTerm: The parts of the algebraic expression joined by
the operations of addition and subtraction are called term.For example : 3x+2 , 2(8y-9) , etc
Polynomial:An algebraic expression in which the exponent of the variable is a non-negative integers is called a polynomial. The highest power of a variable in a polynomial is called the degree of the polynomial.For example :
4x3 + 2x2 – 9x + 8,
Chapter :5
Factoring a Monomial from a polynomial
If a polynomial has only one term, it is called a Monomial.
Monomial
For example : 6x, 8y are monomials.
Chapter :5
Exercise set 5.1 on page=258
Write each number as a product of prime numbers.
2 2002 1002 505 255 5 1
Question :5 Question :6
96482412 6 3 1
222223
Chapter :5
Exercise set 5.1 on page=258
Find the greatest common factor (GCF) for the two give numbers.
Question 11. 72 , 90
723618 9 3 1
22233
904515 5 1
2335
72 = 2x2x2x3x390 = 2x3x3x5GCF= 2X3X3GCF= 18 (ANSWER)
Chapter :5
Exercise set 5.1 on page=258Find the greatest common factor (GCF) for each set of terms.
Question:13 x3 . X . X2
lowest exponent is the GCF= x
Question: 16 6y . 4y2 . 8y3 (2x3)y . (2x2)y.y . (2x2x2)y.y.y GCF= 2y (answer)
Question: 34 x( x + 7) . X + 7 GCF= ( x + 7) (answer)
Chapter :5
Exercise set 5.1 on page=258
Question: 37 3x + 6 = 3x + 3.2 =3(x + 2) GCF= 3 = 3( x + 2) (answer)
Factor the GCF from each term in the expression. If an expression cannot be factored .so state
Question: 41 13x + 5 GCF= Nil
Question: 53 x + 3xy2 = x(1 + 3y2) GCF= x = x(1 + 3y2) answer
Chapter :5
Exercise set 5.1 on page=258
Question: 80 5x(2x – 5) + 3(2x – 5) GCF= (2x – 5) = (2x – 5)(5x + 3) answer
Question: 76 44x5y + 11x3y + 22x2 = (4.11)x2 .x3.y + (11)x2.xy + (2.11)x2
GCF= 11x2
= 11x2 (4x3 . Y + x.y + 2) answer
Note : Remaining Question is your Home Work
Chapter :5
Factoring by Grouping
Let us suppose that : A + B + C + D (So we have four terms here) = (A + B) + (C + D) (Make two groups)
Example:6 on page 261, Factor 4x2 - 2x – 2x + 1 by grouping. = (4x2 - 2x) - (2x - 1) = (2.2.x.x – 2x) – 1( 2x - 1)
= 2x(2x - 1) – 1(2x - 1) = (2x - 1) (2x - 1) answer.
Chapter :5
Exercise set 5.2 on page=263
Factor by groupingQuestion:1 x2 + 4x + 3x + 12 = (x.x + 4x) + (3x + 3.4) = x(x + 4) + 3(x + 4) = (x + 4)(x + 3) answer
Question:12 x2 + 4x + x + 4 = (x.x + 4x) + (x + 4) = x(x + 4) +1(x + 4) = (x + 4)(x + 1) answer
Question:18 35x2 + 21x - 40x - 24 = (7.5x.x + 7.3x) - (8.5x + 8.3) = 7x(x + 3) - 8(x + 3) = (x + 3)(7x - 8) answer
Question:28 12x2 - 9xy + 4xy – 3y2
= 3x(4x - 3y) + y(4x - 3y) = (3x + y) (4x – 3y) answer
Chapter :5
Exercise set 5.2 on page=263
Question:40 x3 - 3x2 + 2x - 6 = x2(x- 3) + 2(x - 3) = (x - 3) (x2 + 2) answer
Question:50 18x2 +27xy + 12xy + 18y2
= 9x(2x + 3y) + 6y(2x + 3y) = (2x + 3y) (9x + 6y) answer
Note : Remaining Question is your home work
Chapter :5
Factoring Trinomials with a = 1
Let us suppose that, ax2 + bx + c where a = 1
Trinomial = Three terms of polynomial
Chapter :5
Exercise set 5.3 on page=272
Your Text
Your Text
Factor the problems. If an expression cannot be factored by a method presented in the section .so state
Question:2 x2 - 7x + 12 = x2 – 3x – 4x + 12 = x( x - 3 )- 4(x - 3) = (x - 3)(x - 4)answer
Question:3 x2 + 6x + 8 = x2 + 2x + 4x + 8 = x( x + 2 )+ 4(x + 2) = (x + 2)(x + 4)answer
Question:6 x2 - x - 6 = x2 – 3x + 2x - 6 = x( x - 3)+ 2(x - 3) = (x - 3)(x + 2)answer
Question:11 x2 + x - 6 = x2 + 3x - 2x - 6 = x( x + 3)- 2(x + 3) = (x + 3)(x - 2)answer
Chapter :5
Exercise set 5.3 on page=273
Question:18 x2 - 10x + 25 = x2 – 5x - 5x + 25 = x( x - 5) - 5(x - 5) = (x - 5)(x - 5)answer
Question:23 x2 + 4x + 4 = x2 + 2x + 2x + 4 = x(x + 2)+ 2(x + 2) = (x + 2)(x + 2)answer
Question:30 x2 - 11x + 10 = x2 – 10x - x + 10 = x( x - 10)- 1(x - 10) = (x - 10)(x - 1)answer
Question:33 x2 - x - 20 = x2 – 5x + 4x - 20 = x( x - 5)+ 4(x - 5) = (x - 5)(x + 4)answer
Chapter :5
Exercise set 5.3 on page=273
TEXT TEXT TEXT TEXT
Question:41 x2 - 2x - 80 = x2 – 10x + 8x - 80 = x( x - 10)+ 8(x - 10) = (x - 10)(x + 8)answer
Question:48 x2 - 6xy + 8y2
= x2 – 4xy - 2xy + 8y2
= x( x – 4y)- 2y(x – 4y) = (x – 4y)(x – 2y)answer
Question:49 x2 + 8xy + 15y2
= x2 + 5xy + 3xy + 15y2
= x( x + 5y) + 3y(x + 5y) = (x + 5y)(x + 3y)answer
Question:50 x2 - 5xy - 14y2
= x2 – 7xy + 2xy - 14y2
= x( x – 7y)+ 2y(x – 7y) = (x – 7y)(x + 2y)answer
Chapter :5
Exercise set 5.3 on page=273
Question:51 2x2 - 12x + 10 =2(x2 – 6x + 5)
=2(x2 – 5x – 1x + 5) =2{ x(x - 5) – 1(x - 5) } =2( x - 5)(x - 1) answer
Question:60 3x3 - 36x2 + 33x =3x(x2 – 12x + 11)
=3x(x2 – 11x – 1x + 11) =3x{ x(x - 11) – 1(x - 11) } =3x( x - 11)(x - 1) answer
Remaining Question is your Home Work
Chapter :5
Exercise set 5.4 on page=286
Factoring Trinomials with a 1Factor completely. If an expression cannot be factored , so state
Question:1 2x2 + 7x + 6 =2x2 + 4x + 3x + 6
=2x(x + 2) + 3(x + 2) =(x + 2)(2x + 3) answer
Question:4 5x2 + 13x + 6 =5x2 + 10x + 3x + 6
=5x(x + 2) + 3(x + 2) =(x + 2)(5x + 3) answer
Question:10 4x2 - 11x + 7 =4x2 - 4x - 7x + 7
=4x(x - 1) - 7(x - 1) =(x - 1)(4x - 7) answer
Chapter :5
Exercise set 5.4 on page=286
Question:22 7x2 + 43x + 6 =7x2 + 42x + x + 6
=7x(x + 6) + 1(x + 6) =(x + 6)(7x + 1) answer
Question:46 36x2 - 36x + 9 =36x2 - 18x - 18x + 9
=18x(2x - 1) - 9(2x - 1) =(18x - 1)(18x - 9) answer
Remaining Question is your Home Work
Chapter :5
Special Factoring Formulas
1.Factor the difference of two squares a2 – b2
2.Factor the sum and difference of two squares a3 + b3 = (a + b)(a2 – ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
Chapter :5
Exercise set 5.5 on page=293
Factor the difference of two squares.Question:1 x2 - 4 =(x)2 – (2)2
=(x + 2)(x - 2) answer
Question:5 x2 - 49 =(x)2 – (7)2
=(x + 7)(x - 7) answer
Question:11 64a2 – 36b2
=(8a)2 – (6b)2
=(8a + 6b)(8a – 6b) answer
Question:12 100x2 – 81y2
=(10x)2 – (9y)2
=(10x + 6y)(10x – 9y) answer
Chapter :5
Exercise set 5.5 on page=293
Question:23 16x2 – 100y4
=(4x)2 – (10y2)2
=(4x + 10y2)(4x – 10y2) answer
Question:24 27x2 – 3y2
=3(9x2 – y2) =3{(3x)2 – (y)2}
=3(3x + y)(3x - y) answer
Chapter :5
Exercise set 5.5 on page=293
Question:29 x3 + 8 =(x)3 + (2)3
=(x + 2)(x2 - 2x + 22) =(x + 2)(x2 – 2x + 4) answer
Factor the sum or difference of to cubes
Question:30 x3 - 8 =(x)3 - (2)3
=(x - 2)(x2 + 2x + 22) =(x - 2)(x2 - 2x + 4) answer
Chapter :5
Exercise set 5.5 on page=293
Question:35 8x3 + 27 =(2x)3 + (3)3
=(2x + 3){(2x)2 - (2x)(2) + (3)2} =(2x + 3)(4x2 - 4x + 9) answer
Question:35 64x3 - 27y3
=(4x)3 - (3y)3
=(4x - 3){(4x)2 + (4x)(3y) + (3y)2} =(4x - 3)(16x2 + 12x + 9y2) answer
Chapter :5
Exercise set 5.5 on page=293
Factor CompletelyQuestion:43 2x2 – 2x - 12 =2x2 - 6x + 4x - 12
=(2x2 - 6x) + (4x - 12) =2x(x - 3) + 4(x - 3) = (x - 3)(2x + 4) answer
Special One
Chapter :5
Exercise set 5.5 on page=293
Question:47 3x2 + 6x + 3 =3x2 + 3x + 3x + 3
=(3x2 + 3x) + (3x + 3) =3x(x + 1) + 3(x + 1) = (x + 1)(3x + 3) answer
Question:63 3x3 – 10x2 – 8x =x(3x2 – 10x – 8)
=x(3x2 – 12x + 2x – 8 ) =x{ (3x2 – 12x) + 2(x – 4)} =x{3x(x – 4) +2(x – 4)} = x (x – 4)(3x + 2) answer
Chapter :5
Assignments (1).
4
7
Question; (1) {-6 , 7 , 12.4 , -9/5 , ¼ , 3 , 0 , 9 , 7 , 0.23 , }
Question (2){-6/3 , 0 , -12 , -19/5 , 1/ 5 , 17 , 5 , 10/5 , , 123 , }3
List those numbers that are:(a). Positive integers.(b). Whole numbers.(c). Integers.(d). Rational numbers .(e). Irrational numbers.(f). Real numbers .
Question (3) X2 – 3x – 5 , x = 2 find the value of “x” & also check ?
Question (4) 2x2 – 6 when x = - 4 find the value of “x” &
check ?
Question (4) 6x2 – 3y3 + 4 when x = 3 & y = -2 find the real
number?
Question (5) 1 3 (lowest term) 8 4
3 5_ _ 3 Question (6) 4 3 1(lowest term) 5 4 2
_ 1 +
Question (7) 2 3 3(lowest term) 3 5 12
2 1 +_ Question(8)solve the equation &check 5.76 – 4.24x – 1.9x = 27.864
Question(9)solve the equation& check – 4(x + 2) – 3x = 20
Q(10)solve the equation& check 9(- y + 3) = - 6y + 15 – 3y + 12
Ali Khan 0788-540242
Chapter :5
Assignments (2).Q(11) Find the GCF of expression ? 5x(2x – 5) + 3(2x – 5)
Q(12) Find the GCF of expression ? 3x(4x – 5) + 1(4x – 5)
Q(13) Factor by grouping ? 10x2 – 12xy – 25xy + 30y2
Q(14) Factor by grouping? 12x2 – 9xy + 4xy – 3y2
Q(15) Factor by grouping? 18x2 + 27xy + 12xy + 18y2
Q(16) Factor the problem ? 3x3 – 36x2 + 33x Q(17) Factor the problem ? x2 – 17x + 60
Q(18) Factor the problem ? x2 – 5xy – 14y2
Q(19) Factor the problem ?
x2 + 18x + 32
Q(20) Factor the problem ?
60x2 + 40x + 5
Q(22) Factor the problem ?
15x2 – xy – 6y2
Q(23) Factor the problem ?
15x2 – xy – 6y2
Q(24) Factor the problem ?
60x2 – 125xy + 60y2
Q(25) Factor the difference of the two squares ?
20x2 – 180
Q(26) Factor the difference of the two squares ?
16x2 – 100y4
Q(27) Factor the difference of the two squares ?
27x4 – 3y2
Q(28) Factor the difference of the two squares ?
4x3 – xy2
Q(30) Factor the sum & difference of the two squares ?
27x3 – 64
Q(31) Factor the sum & difference of the two squares ?
8x3 – 27y2
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