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9 . Algebraic Expressions and Identities

Chapter 9 . Algebraic Expressions and Identities

9. Algebraic Expressions and Identities

Exercise 9.1

1. Identify the terms, their coefficients for each of the following expressions.
(i)

(ii)

(iii)

(iv)

(v)

(vi)

Solution: We have ,

Terms

Coefficients

(i)

5

3

(ii)

1

1

1

1

(iii)

4

– 4

1

(iv)

3

3

– 1

1

– 1

(v)

– 1

(vi)

0.3

– 0.6

0.5

2. Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories?

, 1000 ,

,

,

,

,

,

,

,

,

,

Solution: We have ,

Monomials

Binomials

Trinomials

1000 ,

,

, , ,

,

,

,

The polynomials that do not fit in these categories : and

3. Add the following.
(i)

(ii)

(iii)

(iv)

Solution: (i)

We have ,

Solution: (ii)

We have ,

Solution: (iii)

We have ,

Solution: (iv)

We have ,

4. (a) Subtract :

from

(b) Subtract :

from

(c) Subtract :

from

.

Solution: (a) Subtract : from

(b) Subtract : from

Now,

Solution: (c) Subtract : from .

Now ,

Exercise 9.2

1. Find the product of the following pairs of monomials.
(i)

(ii)

(iii)

(iv)

(v)

Solution: (i)

We have,

(ii)

We have ,

(iii)

We have ,

(iv)

We have,

(v)

We have,

2. Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively.

;

;

;

;

Solution: (i)

Here , length , breadth

We know that ,

Area of rectangle = length × breadth

(ii)

Here , length , breadth

We know that ,

Area of rectangle = length × breadth

(iii)

Here , length , breadth

We know that ,

Area of rectangle = length × breadth

(iv)

Here , length , breadth

We know that ,

Area of rectangle = length × breadth

(v)

Here , length , breadth

We know that ,

Area of rectangle = length × breadth

3. Complete the table of products.

First monomial Second monomial
		…	…	…	…	…
	…		…	…	…	…
	…	…	…	…	…	…
	…	…	......	.......	.....	.......
	…	…	.......	.......	.......	.......
	…	…	.......	.......	......	......

Solution: We have ,

First monomial Second monomial

4. Obtain the volume of rectangular boxes with the following length, breadth and height respectively.
(i)

(ii)

(iii)

(iv)

Solution: (i) Here, Length , Breadth and Height

We know that , the volume of rectangular boxes = length × breadth × height

(ii)

Here, Length , Breadth and Height

We know that , the volume of rectangular boxes = length × breadth × height

(iii)

Here, Length , Breadth and Height

We know that , The volume of rectangular boxes = length × breadth × height

(iv)

Here, Length , Breadth and Height

We know that , the volume of rectangular boxes = length × breadth × height

We have,

5. Obtain the product of
(i)

(ii)

(iii)

(iv)

(v)

Solution: (i)

We have,

(ii)

We have,

(iii)

We have,

(iv)

We have ,

(v)

We have,

Exercise 9.3

1. Carry out the multiplication of the expressions in each of the following pairs.
(i)

(ii)

(iii)

(iv)

(v)

Solution: (i)

We have ,

(ii)

We have,

(iii)

We have ,

(iv)

We have,

(v)

We have ,

2. Complete the table :

First expression

Second expression

Product

(i)

(ii)

(iii)

(iv)

(v)

……..

…….

…….

……

…….

Solution: (i) We have ,

(ii) We have,

(iii) We have,

(iv) We have,

(v) We have,

3. Find the product .

(i)

(ii)

(iii)

(iv)

Solution: (i)

We have ,

(ii)

We have ,

(iii)

We have,

(iv)

We have,

4. (a) Simplify

and find its values for (i)

(ii)

(b) Simplify

and find its value for (i)

, (ii)

(iii)

.

Solution: (a) We have,

(i)

So,

(ii)

So,

5. (a) Add:

and

(b) Add:

and

(c) Subtract:

from

(d) Subtract:

from

.

Solution: (a) Add: and .

We have,

Now ,

(b) Add: and

We have,

and

Now ,

(c) Subtract: from
We have,

and

Now ,

(d) Subtract: from .

We have,

and

Now ,

Exercise 9.4

1. Multiply the binomials.
(i)

and

(ii)

and

(iii)

and

(iv)

and

(v)

and

(vi)

and

Solution: (i) and

We have,

(ii) and

We have,

(iii) and

We have,

(iv) and

We have,

(v) and

We have,

(vi) and

We have,

2. Find the product.
(i)

(ii)

(iii)

(iv)

Solution: (i)

We have,

(ii)

We have,

(iii)

We have,

(iv)

We have,

3. Simplify.
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Solution: (i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

(v) We have,

(vi) We have,

(vii) We have,

(viii) We have,

Exercise 9.5

Standard Identities :

1.

2.

3.

4.

1. Use a suitable identity to get each of the following products.
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

Solution: (i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

(v) We have,

(vi) We have,

(vii) We have,

(viii) We have,

(ix) We have,

(x) We have,

2. Use the identity

to find the following products.
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Solution: (i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

(v) We have,

(vi) We have,

(vii) We have,

3. Find the following squares by using the identities.
(i)

(ii)

(iii)

(iv)

(v)

(vi)

Solution: (i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

(v) We have,

(vi)We have,

4. Simplify.
(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Solution: (i) We have,

(ii)We have,

(iii) We have,

(iv) We have,

(v) We have,

(vi)We have,

(vii) We have ,

5. Show that.
(i)

(ii)

(iii)

(iv)

(v)

Solution: (i)

LHS :

RHS

(ii)

LHS :

(iii)

LHS:

(iv)

LHS:

(v)

LHS :

6. Using identities, evaluate.
(i)

(ii)

(iii)

(iv)

(v)

(vi) 297 × 303 (vii) 78 × 82 (viii)

(ix) 10.5 × 9.5

Solution: (i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

(v) We have,

(vi) We have,

(vii) We have,

(viii) We have,

(ix) We have,

7. Using

, find
(i)

(ii)

(iii)

(iv)

Solution: (i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

8. Using

, find
(i) 103 × 104 (ii) 5.1 × 5.2 (iii) 103 × 98 (iv) 9.7 × 9.8

Solution: (i) We have,

(ii) We have,

(iii) We have,

(iv) We have,