Ncert solutions mathematics class 8 Solutions for Chapter 9 algebraic expressions and identities

Q1) 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

Identify the terms, their coefficients for each of the following expressions.

(i) 5 x y z^{2}-3 z y

(ii) 1+x+x^{2}

(iii) 4 x^{2} y^{2}-4 x^{2} y^{2} z^{2}+z^{2}

(iv) 3 – pq + qr – p

(v) (x/2) + (y/2) – xy

(vi) 0.3a – 0.6ab + 0.5b

Q2) 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

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+5 x, 2 y-3 y^{2}, 2 y-3 y^{2}+4 y^{3}, 5 x-4 y+3 x y, 4 z-15 z^{2}, a b+b c+c d+d a, \mathrm{pqr}, \mathrm{p}^{2} \mathrm{q}+\mathrm{pq}^{2}, 2 \mathrm{p}+2 \mathrm{q}

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}-3 p q+4,5+7 p q-3 p^{2} q^{2}

(iv) l^{2}+m^{2}, m^{2}+n^{2}, n^{2}+l^{2}, 2 l m+2 m n+2 n l

Q3) Add the following:

(i) ab – bc, bc – ca, ca – ab

(ii) a – b + ab, b – c + bc, 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+2n

Q4) (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) Subtract 4a – 7ab + 3b + 12 from 12a – 9ab + 5b – 3

(b) Subtract 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz

(c) Subtract 4 p^{2} q-3 p q+5 p q^{2}-8 p+7 q-10 \text { from } 18-3 p-11 q+5 p q-2 p q^{2}+5 p^{2} q

Find the product of the following pairs of monomials.

(i) 4, 7p

(ii) – 4p, 7p

(iii) – 4p, 7pq

(iv) 4 p^{3},-3 p

(v) 4p, 0

Q1) Find the product of the following pairs of monomials:

(i) 4, 7p

(ii) -4p, 7p

(iii) -4p, 7pq

(iv) 4p^3,\ -3p

(v) 4p, 0

Q2) Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively:

(p,q) ; (10m,5n) ; (20x^2,5y^2) ; (4x,\ 3x^2) ; (3mn,4np)

Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively.

(\mathbf{p}, q) ;(10 \mathrm{m}, 5 \mathrm{n}) ;\left(20 \mathrm{x}^{2}, 5 \mathrm{y}^{2}\right) ;\left(4 \mathrm{x}, 3 \mathrm{x}^{2}\right) ;(3 \mathrm{mn}, 4 \mathrm{np})

Q3) Complete the table of products:

(i)

Complete the following table of products:

Obtain the volume of rectangular boxes with the following length, breadth and height respectively.

(i) 5 a, 3 a^{2}, 7 a^{4}

(ii) 2p, 4q, 8r

(iii) x y, 2 x^{2} y, 2 x y^{2}

(iv) a, 2b, 3c

Q4) Obtain the volume of rectangular boxes with the following length, breadth, and height respectively:

(i) 5a,\ 3a^27a^4

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

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

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

Q5) Obtain the product of:

(i) xy, yz, zx

(ii) a,\ -a^2,a^3

(iii) 2,\ 4y,\ 8y^2,16y^3

(iv) a, 2b, 3c, 6abc

(v) m, -mn, mnp

Obtain the product of

(i) xy, yz, zx

(ii) a,-a^{2}, a^{3}

(iii) 2,4 y, 8 y^{2}, 16 y^{3}

(iv) a, 2b, 3c, 6abc

(v) m, – mn, mnp

Q1) Carry out the multiplication of the expressions in each of the following pairs:

(i) 4p,\ q+r

(ii) ab,\ a\ -\ b

(iii) a+b,\ 7a^2b^2

(iv) a^2-9,\ 4a

(v) pq+qr+rp, 0

Carry out the multiplication of the expressions in each of the following pairs.

(i) 4p, q + r

(ii) ab, a – b

(iii) a + b, 7 a^{2} b^{2}

(iv) a^{2}-9,4 a

(v) pq + qr + rp, 0

Q2) Complete the table:

Complete the table.

Q3) Find the product:

(i)

\left(a^2\right)\times\left(2a^{22}\right)\times\left(4a^{26}\right)

(ii) \left(\frac{2}{3}xy\right)\times\left(\frac{-9}{10}x^2y^2\right)

(iii) \left(\frac{-10}{3}pq^3\right)\times\left(\frac{6}{5}p^3q\right)

(iv) x\times x^2\times x^3\times x^4

Find the product.

(i) a^{2} \times\left(2 a^{22}\right) \times\left(4 a^{26}\right)

(ii) (2 / 3 x y) \times\left(-9 / 10 x^{2} y^{2}\right)

(iii) \left(-10 / 3 \mathrm{pq}^{3} /\right) \times\left(6 / 5 \mathrm{p}^{3} \mathrm{q}\right)

(iv) \text { (x) } \times\left(x^{2}\right) \times\left(x^{3}\right) \times\left(x^{4}\right)

Q4) (a) Simplify 3x\left(4x-5\right)+3 and find values for

(i) x = 3

(ii) x = \frac{1}{2}

(b) Simplify a\left(a^2+a+1\right)+5 and find its value for

(i) a = 0

(ii) a = 1

(iii) a = -1

(a) Simplify 3x (4x – 5) + 3 and find its values for (i) x = 3 (ii) x =1/2

(b) Simplify a\left(a^{2}+a+1\right)+5 and find its value for (i) a = 0, (ii) a = 1 (iii) a = – 1.

(a) Add: p ( p – q), q ( q – r) and r ( r – p)

(b) Add: 2x (z – x – y) and 2y (z – y – x)

(c) Subtract: 3l (l – 4 m + 5 n) from 4l ( 10 n – 3 m + 2 l )

(d) Subtract: 3a (a + b + c ) – 2 b (a – b + c) from 4c ( – a + b + c )

Q5) Add: p\left(p-q\right),\ q\left(q-r\right)\ and\ r\left(r-p\right)

(b) Add: 2x\left(z-x-y\right)and\ 2y\left(z-y-zx\right)

(c) Subtract: 3l\left(l-4m+5n\right)from\ 4l\left(10n-3m+2l\right)

(d) Subtract: 3a\left(a+b+c\right)-2b\left(a-b+c\right)from\ 4c\left(-a+b+c\right)

Q1) Multiply the binomials:

(i) \left(2x+5\right)and\left(4x-3\right)

(ii) \left(y-8\right)\ and\ \left(3y-4\right)

(iii) \left(2.5l-0.5m\right)\ and\ \left(2.5l+0.5m\right)

(iv) \left(a+3b\right)\ and\ \ \left(x+5\right)

(v) \left(2pq+3q^2\right)\ and\ \left(3pq-2q^2\right)

(vi) \left(\frac{3}{4}a^2+3b^2\right)\ and\ 4\left(a^2-\frac{2}{3}b^2\right)

Multiply the binomials.

i) (2x + 5) and (4x – 3)

(ii) (y – 8) and (3y – 4)

(iii) (2.5l – 0.5m) and (2.5l + 0.5m)

(iv) (a + 3b) and (x + 5)

(v) \left(2 p q+3 q^{2}\right) \text { and }\left(3 p q-2 q^{2}\right)

(vi) (3/4 a^2 + 3b^2 ) and 4( a^2 – 2/3 b^2 )

Q2) Find the product:

(i) (5 – 2x)(3 + x)

(ii) (x + 7y)(7x – y)

(iii) \left(a^2+b\right)\left(a+b^2\right)

(iv) \left(p^2-q^2\right)\left(2p+q\right)

Find the product.

(i) (5 – 2x) (3 + x)

(ii) (x + 7y) (7x – y)

(iii) \left(a^{2}+b\right)\left(a+b^{2}\right)

(iv) \left(p^{2}-q^{2}\right)(2 p+q)

Q3) Simplify:

(i) \left(x^2-5\right)\left(x+5\right)+25

(ii) \left(a^2+5\right)\left(b^2+3\right)+5

(iii) \left(t+s^2\right)\left(t^2-s\right)

(iv) \left(a+b\right)\left(c-d\right)+\left(a-b\right)\left(c+d\right)+2\left(ac+bd\right)

(v) \left(x+y\right)\left(2x+y\right)+\left(x+2y\right)\left(x-y\right)

(vi) \left(x+y\right)\left(x^2-xy+y^2\right)

(vii) \left(1.5x-4y\right)\left(1.5x+4y+3\right)-4.5x+12y

(viii) \left(a+b+c\right)\left(a+b-c\right)

Simplify.

(i) \left(x^{2}-5\right)(x+5)+25

(ii) \left(a^{2}+5\right)\left(b^{3}+3\right)+5

(iii) \left(t+s^{2}\right)\left(t^{2}-s\right)

(iv) (a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)

(v) (x + y)(2x + y) + (x + 2y)(x – y)

(vi) (x+y)\left(x^{2}-x y+y^{2}\right)

(vii) (1.5x – 4y)(1.5x + 4y + 3) – 4.5x + 12y

(viii) (a + b + c)(a + b – c)

Use a suitable identity to get each of the following products.

(i) (x + 3) (x + 3)

(ii) (2y + 5) (2y + 5)

(iii) (2a – 7) (2a – 7)

(iv) (3a – 1/2)(3a – 1/2)

(v) (1.1m – 0.4) (1.1m + 0.4)

(vi) \left(a^{2}+b^{2}\right)\left(-a^{2}+b^{2}\right)

(vii) (6x – 7) (6x + 7)

(viii) (- a + c) (- a + c)

(ix) \left(\frac{1}{2} x+\frac{3}{4} y\right)\left(\frac{1}{2} x+\frac{3}{4} y\right)

(x) (7a – 9b) (7a – 9b)

Q1) Use a suitable identity to get each of the following products:

(i) \left(x+3\right)\left(x+3\right)

(ii) \left(2y+5\right)\left(2y+5\right)

(iii) \left(2a-7\right)\left(2a-7\right)

(iv) \left(3a-\frac{1}{2}\right)\left(3a-\frac{1}{2}\right)

(v) (1.1m – 0.4)(1.1m + 0.4)

(vi) \left(a^2+b^2\right)\left(-a^2+b^2\right)

(vii) (6x – 7)(6x + 7)

(viii) (-a + c)(-a + c)

(ix) \left(\frac{x}{2}+\frac{3y}{4}\right)\left(\frac{x}{2}+\frac{3y}{4}\right)

(x) (7a – 9b)(7a – 9b)

Use the identity (x+a)(x+b)=x^{2}+(a+b) x+a b to find the following products.

(i) (x + 3) (x + 7)

(ii) (4x + 5) (4x + 1)

(iii) (4x – 5) (4x – 1)

(iv) (4x + 5) (4x – 1)

(v) (2x + 5y) (2x + 3y)

(vi) \left(2 a^{2}+9\right)\left(2 a^{2}+5\right)

(vii) (xyz – 4) (xyz – 2)

Q2) Using the identity \left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab to find the following products:

(i) \left(x+3\right)\left(x+7\right)

(ii) \left(4x+5\right)\left(4x+1\right)

(iii) \left(4x-5\right)\left(4x-1\right)

(iv) \left(4x+5\right)\left(4x-1\right)

(v) \left(2x+5y\right)\left(2x+3y\right)

(vi) \left(2a^2+9\right)\left(2a^2+5\right)

(vii) \left(xyz-4\right)\left(xyz-2\right)

Q3) Find the following squares by using identities:

(i) \left(b-7\right)^2

(ii) \left(xy+3z\right)^2

(iii) \left(6x^2-5y\right)^2

(iv) \left(\frac{2}{3}m+\frac{3}{2}n\right)^2

(v) \left(0.4p-0.5q\right)^2

(vi) \left(2xy+5y\right)^2

Find the following squares by using the identities.

(i) (b-7)^{2}

(ii) (x y+3 z)^{2}

(iii) \left(6 x^{2}-5 y\right)^{2}

(iv) [(2 m / 3)+(3 n / 2)]^{2}

(v) (0.4 p-0.5 q)^{2}

(vi) (2 x y+5 y)^{2}

Q4) Simplify:

(i) \left(a^2-b^2\right)^2

(ii) \left(2x+5\right)^2-\left(2x-5\right)^2

(iii) \left(7m-8n\right)^2+\left(7m+8n\right)^2

(iv) \left(4m+5n\right)^2+\left(5m+4n\right)^2

(v) \left(2.5p-1.5q\right)^2-\left(1.5p-2.5q\right)^2

(vi) \left(ab+bc\right)^2-2ab^2c

(vii) \left(m^2-n^2m\right)^2+2m^3n^2

Simplify.

(i) \left(a^{2}-b^{2}\right)^{2}

(ii) (2 x+5)^{2}-(2 x-5)^{2}

\text { (iii) }(7 \mathrm{m}-8 \mathrm{n})^{2}+(7 \mathrm{m}+8 \mathrm{n})^{2}

(\mathbf{i v})(4 \mathbf{m}+\mathbf{5} \mathbf{n})^{2}+(5 \mathbf{m}+\mathbf{4} \mathbf{n})^{2}

(v)(2.5 p-1.5 q)^{2}-(1.5 p-2.5 q)^{2}

\text { (vi) }(a b+b c)^{2}-2 a b^{2} c

\text { (vii) }\left(\mathrm{m}^{2}-\mathrm{n}^{2} \mathrm{m}\right)^{2}+2 \mathrm{m}^{3} \mathrm{n}^{2}

Q5) Show that:

(i) \left(3x+7\right)^2-84x=\left(3x-7\right)^2

(ii) \left(9p-5q\right)^2+180pq=\left(9p+5q\right)^2

(iii) \left(\frac{4}{3}m-\frac{3}{4}n\right)^2+2mn=\frac{16}{9}m^2+\frac{9}{16}n^2

(iv) \left(4pq+3q\right)^2-\left(4pq-3q\right)^2=48pq^2

(v) \left(a-b\right)\left(a+b\right)+\left(b-c\right)\left(b+c\right)+\left(c-a\right)\left(c+a\right)=0

Show that.

\text { (i) }(3 x+7)^{2}-84 x=(3 x-7)^{2}

\text { (ii) }(9 p-5 q)^{2}+180 p q=(9 p+5 q)^{2}

\text { (iii) }\left(\frac{4}{3} m-\frac{3}{4} n\right)^{2}+2 m n=\frac{16}{9} m^{2}+\frac{9}{16} n^{2}

(iv)(4 p q+3 q)^{2}-(4 p q-3 q)^{2}=48 p q^{2}

(v)(a-b)(a+b)+(b-c)(b+c)+(c-a)(c+a)=0

Q6) Using identities, evaluate:

(i) 71^2

(ii) 99^2

(iii) 102^2

(iv) 998^2

(v) 5.2^2

(vi) 297\times303

(vii) 78\times82

(viii) 8.9^2

(ix) 1.05\times9.5

Using identities, evaluate.

\text { (i) } 71^{2}

\text { (ii) } 99^{2}

(iii) 102^{2}

\text { (iv) } 998^{2}

\text { (v) } 5.2^{2}

\text { (vi) } 297 \times 303

\text { (vii) } 78 \times 82

\text { (viii) } 8.9^{2}

\text { (ix) } 10.5 \times 9.5

\text { Using } a^{2}-b^{2}=(a+b)(a-b), \text { find }

\text { (i) } 51^{2}-49^{2}

\text { (ii) }(1.02)^{2}-(0.98)^{2}

\text { (iii) } 153^{2}-147^{2}

\text { (iv) } 12.1^{2}-7.9^{2}

Q7) Using a^2-b^2=\left(a+b\right)\left(a-b\right), find

(i) 51^2-49^2

(ii) \left(1.02\right)^2-\left(0.98\right)^2

(iii) 153^2-147^2

(iv) 12.1^2-7.9^2

Q8) Using \left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab, find

(i) 103 x 104

(ii) 5.1 x 5.2

(iii) 103 x 98

(iv) 9.7 x 9.8

\text { Using }(x+a)(x+b)=x^{2}+(a+b) x+a b, \text { find }

(i) 103 x 104

(ii) 5.1 x 5.2

(iii) 103 x 98

(iv) 9.7 x 9.8