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:$1$, $x$and ${x^2}$
Coefficients: 1 in 1, 1 in $x$and 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 in$4{x^2}{y^2}$, - 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 $qr$and -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:$0.3a$,$- 0.6ab$and $0.5b$
Coefficients: 0.3 in $0.3a$, -0.6 in $- 0.6ab$and 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} " />$\eqalign{<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& {\rm{ }}ab{\rm{ }} - {\rm{ }}bc \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& + {\rm{ }}bc{\rm{ }} - {\rm{ }}ca \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& + {\rm{ }} - {\rm{ }}ab{\rm{ }} + {\rm{ }}ca \cr}$
$0$
Hence, addition of $ab - bc,bc - ca,ca - ab$is 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} " />$\eqalign{<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& {\rm{ }}a{\rm{ }} - {\rm{ }}b{\rm{ }} + {\rm{ }}ab \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& + {\rm{ }}b{\rm{ }} - {\rm{ }}c{\rm{ }} + {\rm{ }}bc \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& + {\rm{ }} - {\rm{ }}a{\rm{ }} + {\rm{ }}c{\rm{ }} + {\rm{ }}ac \cr}$
$ab{\rm{ }} + {\rm{ }}bc{\rm{ }} + {\rm{ }}ac$
Hence, addition of $ab - bc,bc - ca,ca - ab$is 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} " />$\eqalign{<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& {\rm{ }}2{p^2}{q^2} - {\rm{ }}3pq{\rm{ }} + {\rm{ }}4 \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& + {\rm{ }} - 3{p^2}{q^2} + 7pq{\rm{ }} + {\rm{ }}5 \cr}$
$- {p^2}{q^2} + {\rm{ }}4pq{\rm{ }} + {\rm{ }}9$
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} " />$\eqalign{<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& {\rm{ }}{l^2} + {m^2} \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& + {\rm{ }}{m^2} + {n^2} \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& + {\rm{ }}{l^2}{\rm{ }} + {n^2} \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& + {\rm{ }}2lm + 2mn + 2nl \cr}$
${\rm{ }}2{l^2} + 2{m^2} + 2{n^2} + 2lm + 2mn + 2nl$
Hence, addition of${l^2} + {m^2},{m^2} + {n^2},{n^2} + {l^2},2lm + 2mn + 2nl$is$2{l^2} + 2{m^2} + 2{n^2} + 2lm + 2mn + 2nl$

Q.4 . (a) Subtract $4a - 7ab + 3b + 12$ from $12ab - 9ab + 5b - 3$
(b) Subtract $3xy + 5yz - 7zx$from $5xy - 2yz - 2zx + 10xyz$
(c) Subtract $4{p^2}q - 3pq + 5p{q^2} - 8p + 7q - 10$from $18 - 3p - 11q + 5pq - 2p{q^2} + 5{p^2}q$
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} " />$\eqalign{<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& {\rm{ }}12a{\rm{ }} - {\rm{ }}9ab{\rm{ }} + {\rm{ }}5b{\rm{ }} - {\rm{ }}3 \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& {\rm{ }}4a{\rm{ }} - {\rm{ }}7ab{\rm{ }} + {\rm{ }}3b{\rm{ }} + 12 \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& ( - ){\rm{ }}( + ){\rm{ }}( - ){\rm{ }}( - ) \cr}$
$8a{\rm{ }} - {\rm{ }}2ab{\rm{ }} + {\rm{ }}2b{\rm{ }} - {\rm{ }}15$

(b)
& {\rm{ }}5xy - {\rm{2yz}} - {\rm{2zx + 10xyz}} \cr
& {\rm{ 3xy + 5yz}} - {\rm{7zx}} \cr
& ( - ){\rm{ }}( - ){\rm{ }}( + ){\rm{ }} \cr} " />$\eqalign{<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& {\rm{ }}5xy - {\rm{2yz}} - {\rm{2zx + 10xyz}} \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& {\rm{ 3xy + 5yz}} - {\rm{7zx}} \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& ( - ){\rm{ }}( - ){\rm{ }}( + ){\rm{ }} \cr}$
$2xy - 7yz + 5zx + 10xyz$

(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} " />$\eqalign{<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& {\rm{ }}18 - 3p - 11q + 5pq - 2p{q^2} + 5{p^2}q{\rm{ }} \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& - {\rm{ }}10 - 8p + 7q - 3pq + 5p{q^2} + 4{p^2}q \cr<br /> </span><span style="font-family: 'times new roman', times, serif; font-size: 12pt; color: #000000;">& ( + )( + ){\rm{ }}( - ){\rm{ }}( + ){\rm{ }}( - ){\rm{ }}( - ){\rm{ }} \cr}$
$28 + 5p - 18q + 8pq - 7p{q^2} + {p^2}q$