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• MATHEMATICS DIGITAL

TEXT BOOK

CLASS IX

BINCY S BABY

B. Ed Mathematics

Reg. No : 18014350005

• CONTENT

Chapter 1

Simple triangles

Of the angles and sides1

Of the sides and angles3

Similarity.6

Third way.7

Chapter 2 Polynomials

Polynomial..8

Polynomial peculiarities9

Polynomial multiplication.11

Degree of a polynomial..............12

Meaning of remainder................13

Doing Division............................13

• Chapter 1

SIMILAR TRIANGLES

OF ANGLES AND SIDES

Here a, b, c are the lengths of the sides of

a triangle and x, y, z are the lengths of the sides

of another triangle with the same angles.

We started with the triangles ABC and

XYZ in which A=X, B=Y, C=Z.

The numbers pairs (x, a), (y, b), (z, c) are

the lengths of the sides opposite the pairs of

equal angles.

Z Y

X

y

x

z b

a

c

C B

A

1

• Now

is the number which shows how

many times the number x is the number a (or

what part of x is a).The other numbers

and

have similar meanings.

Whatever times (or part) we take to change

the side of length a to the side of length x,

such times(or part) is to be taken to change the

side of length b to the side of length y and the

side of length c to the side of length Z.

In other words, all these pairs of lengths

are in the same ratio. So, we can state this result

about the angles lengths of the sides of a

triangle with the same angles, as follows:

If all the angles of a triangle are equal to

the angles of another triangle, then all the pairs

of sides opposite equal angles have the same

ratio.

2

• OF SIDES AND ANGLES

Does the pair of triangles with proportional

sides also have the same angles?

Lets consider the triangle shown below

We have

=

=

We want to check whether the triangles

have the same angles.

For that, what we do is draw a third triangle

with the same angles of ABC, and which is

congruent to XYZ.

b

a

c

C B

A

y

x

z

Z Y

X

3

• In PQR,

OR = x, Q=B, R=C

So, we must also have P=A. Thus the angles

of PQR are equal to the angle of ABC.

So, by the theorem we have proved,

=

=

We actually draw PQR with QR = x, So,

b

a

c

C B

A

y

x

z

R Q

P

y

x

z

Z Y

X

4

• =

=

We started with

=

=

Comparing the last two questions, we get

=

and

=

From these, we get

RP = y and PQ = z

b

a

c

C B

A

y

x

z

R Q

P

y

x

z

Z Y

X

5

• Thus the three sides of PQR are equal to the

three sides of XYZ and so these triangles are

congruent. So, the angles opposite their pairs of

equal sides are also equal:

X=P, Y=Q, Z=R

P=A, Q=B, R=C

Thus we have

X=A, Y=B, Z=C

How do we state the result just proved?

If the sides of a triangle are proportional to the

sides of another triangle, then the angles

opposite such sides are equal

SIMILARITY

If the angles of a triangle are all equal to the

angles of another triangle, then the sides of the

6

• two angles are proportional: and on the

otherhand, if the lengths of the sides of another

triangle, then the angles of one triangle are equal

to the angles of the other.

Two triangles related in this way are said to

be similar.

THIRD WAY

We have seen two ways to show that two

triangles are similar: either proves that they have

the same angles or prove that their sides are

proportional.

There is a third way; if a pair of sides is

proportional and their included angles are equal,

then also the triangles are similar.

if two sides of a triangle are proportional

to two sides of another triangle and if their

included angles are equal, then the triangles are

similar.

7

• Chapter 2

POLYNOMIALS

POLYNOMIALS

Various positive integral powers of x are

multiplied by specific numbers (if may be by 1

also), the products are added or subtracted and

finally a specific number is added or subtracted

(it may be 0) such algebraic expression are

called polynomials.

A polynomial is usually written with the

exponents of x in descending order, the number

without x at the end.

Thus for example, 35x-24x2+4x3 is usually

written as 4x3-24x2+35x; and 15+3x as 3x+15.

8

• POLYNOMIAL PECULARITIES

Look at 2x3-3x2+x-5. We can write it as

2x3+(-3)x2+x+(-5)

Each such addend is called a term of the

polynomial.

Thus in the polynomial we considered just

now, the terms are 2x3, -3x2, x, -5.

Now lets look at the terms themselves.

Apart from the last term, all others are got by

multiplying (Positive integral) powers of x by

specific numbers. Each such multiplier is called

the Co-efficient of the corresponding power.

Thus in our example, the coefficient of x3

is 2, the coefficient of x2 is -3 and the coefficient

of x is 1.

Thus the number added at the end is called

the constant term.

9

Lets simplify (2x+3) + (4x+6)

(2x+3) + (4x+6) = 2x+3 + 4x+6

= (2x+ 4x) + (3 +6)

= 6x+9

(2x +3) - (4x+6) = 2x+3 - 4x-6

= (2x- 4x)+(3 -6)

= - 2x-3

(4x2+3x+1)+(2x2+5x-2)

= (4+2) x2+ (3+5) x+(1-2)

= 6x2+8x-1

(4x2+3x+1) - (2x2+5x-2)

= (4 - 2)x2+(3 - 5)x+(1 + 2)

= 2x2 - 2x + 3

10

• POLYNOMIAL MULTIPLICATION

(x+y) (u+v) = xu+xv+yu+yv

ie, Each number in the first sum should be

multiplied by each number in the second sum

and all these products should be added.

(2x+5)(4x+3

= (2x4x) + (2x 3) + (54x) + (53)

= 8x2+6x+20x+15

= 8x2+26x+15

We can see that for any 3 polynomials

p(x), q(x), r(x).

p(x)r(x)+q(x)r(x) = (p(x)+q(x))r(x)

11

• Example:

(2x+3)(x+1)+( x+4)(x+1)

=((2x+3)+( x+4)) (x+1)

= (3x+7) (x+1)

= 3x2+10x+7

DEGREE OF POLYNOMIAL

The polynomial P(x) =6x4+4x3+5x+1.The

power of x in it are x4, x3, x. The largest

exponent among them is 4. It is called the

degree of the polynomial.

A polynomial whose degree is 1 is called a

first degree polynomial. A polynomial whose

degree is 2 is called a second degree polynomial

and so on.

12

• MEANING OF REMAINDER

In general, when a polynomial cannot be

completely divided by another polynomial, the

remainder should be a polynomial of degree less

than that of the divisor, or a number.

DOING DIVISION

We want to find the quotient and reminder

when 4x2+5 is divided by 2x+3.As taking the

quotient as ax+b and the remainder as c, we get

here

(ax+b)(2x+3)+c = 4x2+5

ie, 2ax2+(3a+2b)x+(3b+c) = 4x2+5

Therefore, 2a= 4, 3a+2b=0, 3b+c=5

ie, 2a=4 ; a=2

For a=2 and 3a+2b=0.We must have 2b= -3a =-6

ie, b= -3 and for b= -3 and 3b+c=5,

Therefore, c=5-3b=14

Thus the quotient is (2x-3) and the remainder 14

13