# NCERT solutions for Exercise 9.1 Class 8 Maths Chapter 9 Algebraic Expressions And Identities

1.Identify the terms, their coefficients for each of the following expressions :
(i) $$5xyz^2 – 3zy$$

(ii) $$1 + x + x^2$$

(iii) $$4x^2y^2 – 4x^2y^2z^2 + z^2$$

(iv) $$3 – pq + qr -rp$$

(v) $$\frac { x }{ 2 } +\frac { y }{ 2 } -xy$$

(vi)$$0.3a – 0.6ab + 0.5b$$

(i) We have the expression $$5xyz^2 – 3zy$$, the terms are $$5xyz^2$$ and $$-3zy$$.
Coefficient of $$xyz^2$$ in the term $$5xyz^2$$ is 5.

Coefficient of $$zy$$ in the term $$– 3yz$$ is $$– 3$$.

(ii)We have the expression $$1 + x + x^2$$, the terms are 1, x and $$x^2$$.

Coefficient of the term 1 is 1.

Coefficient of x in the term x is 1.

Coefficient of $$x^2$$ in the term $$x^2$$ is 1.

(iii) We have the expression $$4xy – 4xyz + z$$ , the terms are $$4x^2y^2, – 4x^2y^2z^2$$ and $$z^2$$.

Coefficient of $$x^2y^2$$ in the term $$4x^2y^2$$ is 4.

Coefficient of $$x^2y^2z^2$$ in the term $$– 4x^2y^2z^2$$ is – 4.

Coefficient of $$z^2$$ in the term $$z^2$$ is 1.

(iv) We have the expression $$3 – pq + qr – rp$$, the terms are 3, – pq, qr and – rp.

Coefficient of the term 3 is 3.

Coefficient of pq in the term – pq is -1.

Coefficient of qr in the term qr is 1.

Coefficient of rp in the term – rp is -1.

(v) We have the expression $$\frac { x }{ 2 } +\frac { y }{ 2 } -xy$$, the terms are $$\frac { x }{ 2 } ,\frac { 7 }{ 2 }$$ and $$– xy$$.

Coefficient of x in the term $$\frac { x }{ 2 }$$ is $$\frac { 1 }{ 2 }$$

Coefficient of y in the term $$\frac { y }{ 2 }$$ is $$\frac { 1 }{ 2 }$$

Coefficient of xy in the term $$– xy$$ is -1.

(vi) In the expression $$0.3a -0.6 ab + 0.5 b$$, the terms are $$0.3a, – 006ab$$and $$0.5b$$.

Coefficient of a in the term 0.3a is 0.3.

Coefficient of ab in the term – 0.6ab is – 0.6.

Coefficient of b in the term 0.5b is 0.5.

2.Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories? $$x + y, 1000, x + x^2 + x^3 + x^4, 7 + y + 5x, 2y – 3y^2, 2y – 3y^2 + 4y^3$$,$$5x – 4y + 3xy, 4z – 15z^2, ab + bc+cd + da, pqr, p^2q + pq^2, 2p + 2q.$$

The calssification of the given expressions are:

Monomials : 1000, pqr

Binomials : $$x + y, 2y – 3y^2, 4z -15z^2, p^2q + pq^2, 2p + 2q.$$

Trinomials : $$7 + y + 5x, 2y – 3y^2 + 4y^3, 5x – 4y + 3x.$$

And the polynomials which don't have any category:

$$x + x^2 + x^3 + x^4, ab + be + cd + da.$$

(i) $$ab – be, be – ca, ea – ab$$

(ii)$$a – b + ab, b-c + be, c – a + ac$$

(iii) $$2p^2q^2 – 3pq + 4, 5 + 7pq – 3p^2q^2$$

(iv)$$l^2 + m^2, m^2 + n^2, n^2 + l^2, 2lm + 2mn + 2nl$$

(i) We have the expressions as:$$ab – bc, bc – ca, ca – ab$$

$$(ab – bc) + (bc – ca) + (ca – ab)$$

$$\Rightarrow ab – bc + bc – ca + ca – ab$$

$$\Rightarrow (ab – ab) + (bc – bc) + (ca – ca)\quad$$ [Placing the like terms together]

$$\Rightarrow 0 + 0 + 0$$

$$\Rightarrow 0$$

(ii) We have the expressions as:$$a – b + ab, b – c + bc, c – a + ac$$

$$(a – b + ab) + (b – c + bc) + (c – a + ac)$$

$$\Rightarrow a – b + ab + b – c + bc + c – a + ac$$

$$\Rightarrow (a – a) + (b – b) + (c – c) + ab + bc + ac\quad$$[Placing all the like terms together]

$$\Rightarrow 0 + 0 + 0 + ab + bc + ac$$

$$\Rightarrow ab + bc + ac$$

(iii)We have the expressions as:$$2p^2q^2 – 3pq + 4, 5 + 7pq – 3p^2q^2$$

Adding these we get: $$2p^2q^2 – 3pq + 4 + 5 + 7pq – 3p^2q^2$$

$$\Rightarrow (2p^2q^2– 3p^2q^2)+(– 3pq+ 7pq)+ (4 + 5)\quad$$[Placing like terms together]

$$\Rightarrow -p^2q^2 + 4pq+ 9$$

(iv)We have the expressions as:$$l^2 + m^2, m^2 + n^2, n^2 + l^2, 2lm + 2mn + 2nl$$
Adding these we get: $$(l^2 + m^2)+( m^2 + n^2)+( n^2 + l^2)+ (2lm + 2mn + 2nl)$$

$$\Rightarrow (l^2+l^2)+ (m^2+m^2)+ (n^2+n^2)+(2lm + 2mn + 2nl)\quad$$[Placing like terms together]

$$\Rightarrow (2l^2)+ (2m^2)+ (2n^2)+(2lm + 2mn + 2nl$$

$$\Rightarrow 2(l^2+ m^2+n^2+lm + mn + nl)$$

4.(a) Subtract $$4a – 7ab + 3b + 12$$ from $$12a – 9ab + 5b – 3$$
(b) Subtract $$3xy + 5yz – 7zx$$ from $$5xy – 2yz – 2zx + 10xyz$$

(c) Subtract $$4p^2q – 3pq + 5pq^2 – 8p + 7q – 10$$ from $$18 – 3p – 11q + 5pq – 2pq^2 + 5p^2q$$

(a)We have the expressions to subtract:$$4a – 7ab + 3b + 12$$ from $$12a – 9ab + 5b – 3$$
Subtracting these we get: $$(12a – 9ab + 5b – 3)-(4a – 7ab + 3b + 12 )$$

$$(12a – 9ab + 5b – 3)+(-4a +7ab - 3b - 12 )\quad$$[Taking (-) inside the second bracket]

$$\Rightarrow (12a-4a)+ (– 9ab+7ab)+ (+ 5b- 3b )+( – 3 - 12 )\quad$$[Placing like terms together]

$$\Rightarrow (8a)+ (-2ab)+ (2b)+(-15)$$

$$\Rightarrow (8a-2ab+ 2b-15)$$

(b)We have the expressions to subtract:$$3xy + 5yz – 7zx$$ from $$5xy – 2yz – 2zx + 10xyz$$
Subtracting these we get: $$(5xy – 2yz – 2zx + 10xyz)-(3xy + 5yz – 7zx)$$

$$(5xy – 2yz – 2zx + 10xyz)+(-3xy - 5yz + 7zx )\quad$$[Taking (-) inside the second bracket]

$$\Rightarrow (5xy-3xy )+ (– 2yz - 5yz)+ (– 2zx+ 7zx )+( + 10xyz )\quad$$[Placing like terms together]

$$\Rightarrow (2xy)+ (-7yz)+ (5zx)+(10xyz)$$

$$\Rightarrow (2xy-7yz+5zx+10xyz)$$

(c)We have the expressions to subtract:$$4p^2q – 3pq + 5pq^2 – 8p + 7q – 10$$ from $$18 – 3p – 11q + 5pq – 2pq^2 + 5p^2q$$
Subtracting these we get: $$(18 – 3p – 11q + 5pq – 2pq^2 + 5p^2q )-(4p^2q – 3pq + 5pq^2 – 8p + 7q – 10)$$

$$\Rightarrow(18 – 3p – 11q + 5pq – 2pq^2 + 5p^2q)+(-4p^2q +3pq-5pq^2 +8p-7q+ 10)\quad$$[Taking (-) inside the second bracket]

$$\Rightarrow (+18+10)+(-3p+8p)+(-11q-7q)+(5pq+3pq)+(-2pq^2-5pq^2)+(+ 5p^2q-4p^2q) \quad$$[Placing like terms together]

$$\Rightarrow (+28)+ (+5p)+(-18q)+(8pq)+(-7pq^2)+(+p^2q)$$

$$\Rightarrow (+28+5p-18q+8pq-7pq^2+p^2q)$$

4.Obtain the volume of rectangular boxes with the following length, breadth and height respectively.
(i) $$5a, 3a^2, 7a^4$$

(ii) $$2p, 4q, 8r$$

(iii) $$xy, 2x^2y, 2xy^2$$

(iv) $$a, 2b, 3c$$

(i)Given that, length = 5a, breadth = $$3a^2$$, height = $$7a^4$$

∴Volume of the box =$$l \times b \times h = 5a \times 3a^2 \times 7a^4 = 105 a^7 \;$$cu. units

(ii)Given that, length = 2p, breadth = 4q, height = 8r

∴Volume of the box =$$l \times b \times h = 2p \times 4q \times 8r = 64pqr\;$$ cu. units

(iii)Given that, length = xy, breadth =$$2x^2y$$, height = 2xy^2\)

Volume of the box =$$l \times b \times h = xy \times 2x^2y \times 2xy^2 = (1 \times 2 \times 2) \times xy \times x^2y \times xy^2 = 4x^4y^4 \;$$cu. units

(iv)Given that, length = a, breadth = 2b, height = 3c

∴Volume of the box = $$l \times b \times h = a \times 2b \times 3c = (1 \times 2 \times 3)abc = 6 abc\;$$ cu. units