Chapter No 5

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


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


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


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


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


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


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


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


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


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


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


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


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



=(4x)2 – (10y2)2

=(4x + 10y2)(4x – 10y2) answer


=3(9x2 – y2) =3{(3x)2 – (y)2}

=3(3x + y)(3x - y) answer

Chapter :5


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


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


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


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


60x2 + 40x + 5


15x2 – xy – 6y2


15x2 – xy – 6y2


60x2 – 125xy + 60y2

Q(25) Factor the difference of the two squares ?

20x2 – 180


16x2 – 100y4


27x4 – 3y2


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

Documents

Chapter No 5