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Algebraic Expressions And Identities : Exercise 9.1 (Mathematics NCERT Class 8th)


Q.1 Identify the terms, their coefficients for each of the following expressions.
(i) 5xy{z^2} - 3zy (ii) 1 + x + {x^2} (iii) 4{x^2}{y^2} - 4{x^2}{y^2}{z^2} + {z^2}
(iv) 3 - pq + qr - rp (v) {x \over 2} + {y \over 2} - xy (vi) 0.3a - 0.6ab + 0.5b
Sol. (i) 5xy{z^2} - 3zy
Terms: 5xy{z^2}and  - 3zy
Coefficients: 5 in 5xy{z^2}and -3 in  - 3zy

(ii) 1 + x + {x^2}
Terms:1, xand {x^2}
Coefficients: 1 in 1, 1 in xand 1 in {x^2}

(iii) 4{x^2}{y^2} - 4{x^2}{y^2}{z^2} + {z^2}
Terms:4{x^2}{y^2}, - 4{x^2}{y^2}{z^2}and{z^2}
Coefficients: 4 in4{x^2}{y^2}, - 4 in  - 4{x^2}{y^2}{z^2}and 1 in {z^2}

(iv) 3 - pq + qr - rp
Terms: 3,  - pq, qr and  - rp
Coefficients: 3 in 3. -1 in  - pq, 1 in qrand -1 in  - rp

(v) {x \over 2} + {y \over 2} - xy
Terms: {x \over 2}, {y \over 2}and  - xy
Coefficients: {1 \over 2}in {x \over 2}, {1 \over 2} in {y \over 2}and -1 in  - xy

(vi) 0.3a - 0.6ab + 0.5b
Terms:0.3a, - 0.6aband 0.5b
Coefficients: 0.3 in 0.3a, -0.6 in  - 0.6aband 0.5 in 0.5b

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

x + y,1000,x + {x^2} + {x^3} + {x^4},7 + y + 5x,2y - 3{y^2},2y - 3{y^2} + 4{y^3},5x - 4y + 3xy,4z - 15{z^2}
ab + bc + cd + da,pqr,{p^2}q + p{q^2},2p + 2q
Sol. The classification of polynomials as monomials, binomials, trinomials are as follows:
Monomials:1000,pqr
Binomials: x + y,2y - 3{y^2},4z - 15{z^2},{p^2}q + p{q^2},2p + 2q
Trinomials: 7 + y + 5x,2y - 3{y^2} + 4{y^3},5x - 4y + 3xy
Polynomials that do not fit in any of these categories:
x + {x^2} + {x^3} + {x^4},ab + bc + cd + da

Q.3 Add the following.
(i) ab - bc,bc - ca,ca - ab
(ii)
a - b + ab,b - c + bc,c - a + ac
(iii) 2{p^2}{q^2} - 3pq + 4,5 + 7pq - 3{p^2}{q^2}
(iv) {l^2} + {m^2},{m^2} + {n^2},{n^2} + {l^2},2lm + 2mn + 2nl
Sol. (i) ab - bc,bc - ca,ca - ab
& {\rm{ }}ab{\rm{ }} - {\rm{ }}bc \cr
& + {\rm{ }}bc{\rm{ }} - {\rm{ }}ca \cr
& + {\rm{ }} - {\rm{ }}ab{\rm{ }} + {\rm{ }}ca \cr} " />
0
Hence, addition of ab - bc,bc - ca,ca - abis 0.

(ii) a - b + ab,b - c + bc,c - a + ac
& {\rm{ }}a{\rm{ }} - {\rm{ }}b{\rm{ }} + {\rm{ }}ab \cr
& + {\rm{ }}b{\rm{ }} - {\rm{ }}c{\rm{ }} + {\rm{ }}bc \cr
& + {\rm{ }} - {\rm{ }}a{\rm{ }} + {\rm{ }}c{\rm{ }} + {\rm{ }}ac \cr} " />
ab{\rm{ }} + {\rm{ }}bc{\rm{ }} + {\rm{ }}ac
Hence, addition of ab - bc,bc - ca,ca - abis ab + bc + ac.

(iii) 2{p^2}{q^2} - 3pq + 4,5 + 7pq - 3{p^2}{q^2}
& {\rm{ }}2{p^2}{q^2} - {\rm{ }}3pq{\rm{ }} + {\rm{ }}4 \cr
& + {\rm{ }} - 3{p^2}{q^2} + 7pq{\rm{ }} + {\rm{ }}5 \cr} " />
 - {p^2}{q^2} + {\rm{ }}4pq{\rm{ }} + {\rm{ }}9
Hence, addition of 2{p^2}{q^2} - 3pq + 4,5 + 7pq - 3{p^2}{q^2}is  - {p^2}{q^2} + 4pq + 9.

(iv) {l^2} + {m^2},{m^2} + {n^2},{n^2} + {l^2},2lm + 2mn + 2nl
& {\rm{ }}{l^2} + {m^2} \cr
& + {\rm{ }}{m^2} + {n^2} \cr
& + {\rm{ }}{l^2}{\rm{ }} + {n^2} \cr
& + {\rm{ }}2lm + 2mn + 2nl \cr} " />
{\rm{ }}2{l^2} + 2{m^2} + 2{n^2} + 2lm + 2mn + 2nl
Hence, addition of{l^2} + {m^2},{m^2} + {n^2},{n^2} + {l^2},2lm + 2mn + 2nlis2{l^2} + 2{m^2} + 2{n^2} + 2lm + 2mn + 2nl

Q.4 . (a) Subtract 4a - 7ab + 3b + 12 from 12ab - 9ab + 5b - 3
(b) Subtract 3xy + 5yz - 7zxfrom 5xy - 2yz - 2zx + 10xyz
(c) Subtract 4{p^2}q - 3pq + 5p{q^2} - 8p + 7q - 10from 18 - 3p - 11q + 5pq - 2p{q^2} + 5{p^2}q
Sol.
(a)
& {\rm{ }}12a{\rm{ }} - {\rm{ }}9ab{\rm{ }} + {\rm{ }}5b{\rm{ }} - {\rm{ }}3 \cr
& {\rm{ }}4a{\rm{ }} - {\rm{ }}7ab{\rm{ }} + {\rm{ }}3b{\rm{ }} + 12 \cr
& ( - ){\rm{ }}( + ){\rm{ }}( - ){\rm{ }}( - ) \cr} " />
8a{\rm{ }} - {\rm{ }}2ab{\rm{ }} + {\rm{ }}2b{\rm{ }} - {\rm{ }}15

(b)
& {\rm{ }}5xy - {\rm{2yz}} - {\rm{2zx + 10xyz}} \cr
& {\rm{ 3xy + 5yz}} - {\rm{7zx}} \cr
& ( - ){\rm{ }}( - ){\rm{ }}( + ){\rm{ }} \cr} " />
2xy - 7yz + 5zx + 10xyz

(c)
& {\rm{ }}18 - 3p - 11q + 5pq - 2p{q^2} + 5{p^2}q{\rm{ }} \cr
& - {\rm{ }}10 - 8p + 7q - 3pq + 5p{q^2} + 4{p^2}q \cr
& ( + )( + ){\rm{ }}( - ){\rm{ }}( + ){\rm{ }}( - ){\rm{ }}( - ){\rm{ }} \cr} " />
28 + 5p - 18q + 8pq - 7p{q^2} + {p^2}q



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