Chapter 9: Algebraic Expressions and Identities

Exercise 9.1

Q.1. Identify the terms and their coefficients for each of the following expressions.

(I) 5abc2 – 3cb

Terms :  5abc2  

3cb

Coefficients: 5, -3

(II) 1+a+a2

Terms: 1, a, a2

Coefficients: 1, 1, 1

(III) 4x2y– 4 x2y2z+  z2

Terms: 4x2y2   ,  -4 x2y2z2    ,  Z2

Coefficient: 4,  -4,  1

(IV) 3 – xy + yz – zx

Terms:   3:  -xy,  yz,  -zx

Coefficient:  3:  -1,  1,  -1

(V) \(\frac{a}{2}+\frac{b}{2}-ab\)

Terms: \(\frac{a}{2}\),  \(\frac{b}{2}\),  -ab

Coefficient: \(\frac{1}{2}\),  \(\frac{1}{2}\),  -1

(VI)0.3x-0.6xy+0.5y

Terms: 0.3x,  -0.6xy,  0.5y

Coefficient: 0.3,  -0.6,  0.5

 

Q.2. Check whether the following polynomials are monomials, binomials or trinomials. Find out which polynomials do not fit any of these three categories?

1)  x+y,

2)  1000,

3)  \(x+x^{2}+x^{3}+x^{4}\),

4)  7+y+5x,

5)  \(2y-3y^{2}\),

6)  \( 2y-3y^{2}+4y^{3}\),

7)  5x-4y+3xy,

8)  \( 4z-15z^{2}\),

9)  ab+bc+cd+da,

10)  pqr,

11)  \( p^{2}q+pq^{2}\),

12)  2p+2q,

Answer:

Monomials: 1000,   pqr

Binomials: x+y,   \( 2y-3y^{2}\),  \( 4z-15z^{2}\),   \( p^{2}q+pq^{2}\),    2p+2q

Trinomials: 7+y+5x,    \( 2y-3y^{2}+4y^{3}\),      5x-4y+3xy

Polynomials that do not fit any of these categories are :

\(x+x^{2}+x^{3}+x^{4}\), ab+bc+cd+da

 

Q.3.Add the following :

Note: The given expressions written in separate rows, with like terms one below the other and then the addition of these expressions are done.

(I)ab – bc,    bc – ca,    ca – ab

ab-bc

+   bc-ca

+  -ab+ca

=           0

(II) x – y+xy,       y-z+yz,        z-x+xz

x – y + xy

+      y -z+yz

+       -x+z +xz

=        xy+yz+xz

(III) \(2a^{2}b^{2}-3ab+4\, \, \, 5+7ab-3a^{2}b^{2}\)

\(2a^{2}b^{2}-3ab+4\)

+    \(-3a^{2}b^{2}+7ab+5\)

\(-a^{2}b^{2}+4ab+9\)

 

(IV) \(a^{2}+b^{2}\, \, \, b^{2}+c^{2},\, \, \, c^{2}+a^{2},\, \, \, 2ab+2bc+2ca\)

\(a^{2}+b^{2}\)

+     \( b^{2}+c^{2}\)

+     \( c^{2}+a^{2}\)

+     2ab+2bc+2ca

=      \(2a^{2}+2b^{2}+2c^{2}+2ab+2bc+2ca\)

 

Q.4. (i)Substract 4x-7xy+3y+12 from 12x-9xy+5y-3

Answer:

12x – 9xy + 5y –  3

4x – 7xy + 3y + 12

(-)     (+)    (-)    (-)

8x – 2xy + 2y – 15

(ii)Substract 3xy +5yz -7zx from 5xy-2yz-2zx+10xyz

5xy – 2yz -2zx +10xyz

3xy + 5yz -7zx

(-)      (-)      (+)

2xy-7yz + 5zx  +10xyz

(iii) \(Substract\: 4p^{2}q-3pq+5pq^{2}-8p+7q-10 \:\, from\: \, 18-3p+11q+5pq-2pq^{2}+5p^{2}q\)

\(18-3p-11q+5pq-2pq^{2}+5p^{2}q\)

 

\(-10-8p+7q-3pq+5pq^{2} +4p^{2}q\)

(+)          (+)   (-)  (+)   (-)            (-)

\(28+5p-18q+8pq-7pq^{2} +p^{2}q\)

 

Exercise: 9.2

Q.1.For the following pairs of monomials find the product.

(I)5, 6a

Answer: \(5\times 6\times a\\ =30a\)

(II)-5a, 6 a

Answer: \(-5a\times 6a\times \\ =-5\times a\times 6\times a\\ =(-5\times 6)\times (a\times a)\\ =-30a^{2}\)

(III) )-5a, 6 ab

Answer: \(-5a\times 6ab\times \\ =-5\times a\times 6\times a \times b\\ =(-5\times 6)\times (a\times a\times b)\\ =-30a^{2}b\)

(IV) ) \(5a^{3}\),- 4 a

Answer: \(5a^{3}\times -4a \\ =5\times (-4)\times a\times a\times a \times a\\ =-20a^{4}\)

(V)5a, 0

Answer: \(5a\times 0\\ =5\times a\times 0\\ =0\)

 

Q.2.calculate the area of rectangles.Where the pairs of monomials  are lengths and breadths respectively.

NOTE: area of rectangle =\(length \times breadth\)

  • (a, b)

Area= \(a \times b\\ =ab\)

  • (10a, 5b)

Area = \(10a \times 5b\\ =10\times 5\times a\times b\\ =50ab\)

  • (\(20p^{2},5q^{2}\))

Area = \(20p^{2}\times 5q^{2}\\ =20\times 5\times p^{2}\times q^{2}\\ =100p^{2}q^{2}\)

  • (\(4a,3a^{2}\))

Area = \(4a\times 3a^{2}\\ =4\times 3\times a\times a^{2}\\ =12a^{3}\)

  • (4ab,3bc)

Area= \(4ab\times 3bc\\ =4\times 3\times a\times b\times b\times c\\ =12ab^{2}c\)

 

Q.3.Complete the table of product.

First monomial

Second monomial

 

2x

-5y \(3x^{2}\) -4xy \(7x^{2}y\) \(-9x^{2}y^{2}\)
2x \(4x^{2}\)
-5y \(-15x^{2}y\)
\(3x^{2}\)
-4xy

 

Solution:

First monomial

Second monomial

 

2x

-5y \(3x^{2}\) -4xy \(7x^{2}y\) \(-9x^{2}y^{2}\)
2x \(4x^{2}\) \(-10xy\) \(6x^{3}\) \(-8x^{2}y\) \(14x^{3}y\) \(18x^{3}y^{2}\)
-5y   \(-10xy\) \(-15x^{2}y\) \(-15x^{2}y\) \(20xy^{2}\) \(-35x^{2}y^{2}\) \(45x^{2}y^{3}\)
\(3x^{2}\) \(6x^{3}\) \(-15x^{2}y\) \(9x^{4}\) \(-12x^{3}y\) \(21x^{4}y\) \(-27x^{4}y^{2}\)
-4xy \(-8x^{2}y\) \(20x^{2}y\) \(-12x^{3}y\) \(16x^{2}y^{2}\) \(-28x^{3}y^{2}\) \(36x^{3}y^{3}\)

 

 

Q.4.Rectangular boxes with the length \(,\)breadth \(,\) and height are given respectively. Find the volume.

(I) \(5x, 3x^{2}, 7x^{4}\)

Answer: \(Volume=5x\times 3x^{2}\times 7x^{4}=5\times 3\times 7\times x\times x^{2}\times x^{4}=105x^{7}\)

(II)2p, 4q, 8r

Answer: \(Volume=2p\times 4q\times 8r= 2\times 4\times 8\times p\times q \times r=64pqr\)

(III) \(ab, 2a^{2}b, 2ab^{2}\)

Answer:  \(Volume=ab\times 2a^{2}b\times 2ab^{2}=2\times 2\times ab\times a^{2} b\times ab^{2}=4a^{4}b^{4}\)

(IV)p, 2q, 3r

Answer:  \(Volume=p\times 2q\times 3r= 2\times 3\times p\times q \times r=6pqr\)

 

Q.5.Find the Product of the following:

(I)ab, bc, ca

Answer: \(ab\times bc\times ca\)= \(a^{2}b^{2}c^{2}\)

(II) \(x, -x^{2}, x^{3}\)

Answer: \(x\times (-x^{2})\times x^{3}=-x^{6}\)

(III) \(2, 4a, 8a^{2}, 16a^{3}\)

Answer: \(2\times 4a\times 8a^{2}\times 16a^{3}=1024a^{6}\)

(IV)x, 2y, 3z, 6xyz

Answer: \(x\times 2y\times 3z\times 6xyz=36x^{2}y^{2}z^{2}\)

(V)m, -mn, mnp

Answer: \(m\times -mn\times mnp=-m^{3}n^{2}p\)

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