UNIT 1.6 PART 2

Example 1.6.6

21 1 2 3

3 3 3 3

If 3,

1 is undefined

3

3 is an ext raneous solu t ion

1

x x

x x x x

x

x

S S

: no. of boys

7 : no. of gir ls

x

x

Example 1.6.7

A man gave Php 120M to 7 children,giving Php 60M to the boys and thesame amount to the gir ls. In th is way,each boy received Php 5M more thaneach gir l. F ind the number of boys andgir ls.

60: money in M received by each boy

60: money in M received by each girl

7

60 605

7

: 7

x

x

x x

LCD x x

Example 1.6.7

2

2

60 607 7 5

7

420 60 60 5 7 , 0,7

420 60 60 35 5 , 0,7

5 155 420 0, 0,7

x x x xx x

x x x x x

x x x x x

x x x

Example 1.6.7

2

2

5 155 420 0, 0,7

31 84 0

28 3 0

28 3

There are 3 boys and 4 girls.

x x x

x x

x x

x x

Example 1.6.7

Discriminant

2

2

4

2

Discr iminant : 4

b b acx

a

D b ac

Discriminant

2 4

2

0 : only one rea l solut ion

0 : two dist inct rea l solut ions

0 : two dist inct complex solut ions

b b acx

a

D

D

D

Example 1.6.8

2

2

Find the na ture of the solu t ions

of 4 8 3 0.

4 8 3

4 64 4 4 3 16 0

The equa t ion has 2 dist inct

rea l solu t ions.

x x

a b c

b ac

Sum/Product of Roots

2 2

Sum:

4 4

2 2

2

2

b b ac b b ac

a a

b

a

b

a

Sum/Product of Roots

2 2

22 2

2

2 2

2

Product :

4 4

2 2

4

4

4

4

b b ac b b ac

a a

b b ac

a

b b ac

a

Sum/Product of Roots

2 22

2 2

2

2

4

4

4

4

4

4

b b ac

a

b b ac

a

ac

a

c

a

Sum/Product of Roots

2 0

Sum of Roots:

Product of Roots:

ax bx c

b

a

c

a

Example 1.6.9

2

2

2

Find the sum and product of the

roots of 2 3 4.

2 3 4

2 3 4 0

2 3 4

3 3 4Sum: Product : 2

2 2 2

x x

x x

x x

a b c

Cubic Equations

3 2 0

To solve:

Factor ing

Using quadra t ic formula

ax bx cx d

Example 1.6.10

3 1

1 31

2

1 3 1 31, ,

2 2

x

ix x

i iS S

Division of Polynomials

Long Division

Synthet ic Division

Example 1.6.11

3Divide 3 2 by 2 using

1. long division

2. synthet ic division

x x x

23 2

3 2

2

2

2 1

1. 2 0 3 2

2

2 3

2 4

2

2

4

x x

x x x x

x x

x x

x x

x

x

Quotient

Remainder

3 2

2

2. 2 0 3 2

2 1 0 3 2

2 4 2

1 2 1 4

2 1 4

x x x x

x x r

Division of Polynomials

If a polynomia l is divided

by we get a quot ien t

and a remainder .

P x

x a Q x

R

P x rQ x

x a x a

P x x a Q x r

3

2

32

3 2

3 2 divided by 2

2 1

4

3 2 42 1

2 2

3 2 2 2 1 4

P x x x x

Q x x x

r

x xx x

x x

x x x x x

Remainder Theorem

If a polynomial is divided by ,

the remainder is equal to .

P x x a

P a

Example 1.6.11

3

3

3

Use the remainder theorem to

determine the remainder when

3 2 is divided by 2.

3 2

2 2 3 2 2 8 6 2 4

4 is the remainder .

x x x

P x x x

P

23 2

3 2

2

2

2 1

1. 2 0 3 2

2

2 3

2 4

2

2

4

x x

x x x x

x x

x x

x x

x

x

Quotient

Remainder

Factor Theorem

is a root of the equat ion 0

if and only if is a factor of .

a P x

x a P x

Example 1.6.11

4 3

4 3

4 3

4 3

Use the factor theorem to determine

if 1 is a factor of 2 4 1.

1 1

Is 1 a solu t ion to 2 4 1 0

1 2 1 1 4 1 1 2 1 4 1 0

1 is a factor of 2 4 1.

x x x x

x x

P x x x x

P

x x x x

Example 1.6.11

3 2

3 2

3 2

3 2

3 2

23

Using the Factor Theorem, solve the equa t ion

6 11 6 (Hint : Show tha t 1 is a factor .)

6 11 6

6 11 6 0

Is 1 a factor of 6 11 6 ?

6 11 6

1 1 6 1 11 1 6 0

1 is a factor of

x x x x

x x x

x x x

x x x x

P x x x x

P

x

3 2 6 11 6.x x x

3 2

3 2

2

3 2

2

6 11 6 1

1 6 11 6

1 1 6 11 6

1 5 6

1 5 6 0

5 6 0

6 11 6 0

1 5 6 0

x x x x

x x x x

x x r

x x x

x x x

3 2

2

6 11 6 0

1 5 6 0

1 2 3 0

1 0 2 0 3 0

1 2 3

1,2,3

x x x

x x x

x x x

x x x

x x x

S S