Maths Chapter-09: Algebraic Expressions and Identities
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) 4x2y2 – 4 x2y2z2 + 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) \(Substract\: 4p^{2}q-3pq+5pq^{2}-8p+7q-10 \:\, from\: \, 18-3p+11q+5pq-2pq^{2}+5p^{2}q\)
\(18-3p-11q+5pq-2pq^{2}+5p^{2}q\)
\(-10-8p+7q-3pq+5pq^{2} +4p^{2}q\)
(+) (+) (-) (+) (-) (-)
\(28+5p-18q+8pq-7pq^{2} +p^{2}q\)
Ex: 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: \(m\times -mn\times mnp=-m^{3}n^{2}p\)