ALGEBRAIC EXPRESSIONS AND IDENTITIES

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9 ALGEBRAIC EXPRESSIONS AND IDENTITIES Exercise 9.1 Q.1. Identify the terms, their coefficients for each of the following expressions. (i) 5xyz 3zy (ii) 1 + x + x (iii) 4x y 4x y z + z (iv) 3 pq + qr rp x y xy (vi) 0.3a 0.6ab + 0.5b (v) + Ans. (i) There are two terms; 5xyz and 3zy (ii) There are three terms; 1, x and x (iii) There are three terms; 4x y, 4x y z and z 1

(iv) There are four terms; 3, pq, qr and rp x y (v) There are three terms;, and xy (vi) There are three terms; 0.3a 0.6ab and 0.5b Q.. Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories? x + y, 1000, x + x + x 3 + x 4, 7 + y + 5x, y 3y, y 3y + 4y 3, 5x 4y + 3xy, 4z 15z, ab + bc + cd + da, pqr, p q + pq, p + q Ans. Monomials : 1000, pqr Binomials : x + y, y 3y, 4z 15z, p q + pq, p + q Trinomials : 7 + y + 5x, y 3y + 4y 3, 5x 4y + 3xy Polynomials that do not fit in these categories : x + x + 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) p q 3pq + 4, 5 + 7pq 3p q (iv) l + m, m + n, n + l, lm + mn + nl Ans. (i) (ab bc) + (bc ca) + (ca ab) ab bc + bc ca + ca ab 0 + 0 + 0 0 (ii) (a b + ab) + (b c + bc) + (c a + ac) a b + ab + b c + bc + c a + ac ab + bc + ac (iii) (p q 3pq + 4) + ( 3p q + 7pq + 5) p q 3pq + 4 3p q + 7pq + 5 p q + 4pq + 9 (iv) (l + m ) + (m + n ) + (l + n ) + (lm + mn + nl) l + m + m + n + l + n + lm + mn + nl l + m + n + lm + mn + nl (l + m + n + lm + mn + nl) Q.4. (a) Subtract 4a 7ab + 3b + 1 from 1a 9ab + 5b 3 (b) Subtract 3xy + 5yz 7zx from 5xy yz zx + 10xyz (c) Subtract 4p q 3pq + 5pq 8p + 7q 10 from 18 3p 11q + 5pq pq + 5p q Ans. (a) (1a 9ab + 5b 3) (4a 7ab + 3b + 1) 1a 9ab + 5b 3 4a + 7ab 3b 1 8a ab + b 15 (b) (5xy yz zx + 10xyz) (3xy + 5yz 7zx) 5xy yz zx + 10xyz 3xy 5yz + 7zx xy 7yz + 5zx + 10xyz 3

(c) (18 3p 11q + 5pq pq + 5p q) (4p q 3pq + 5pq 8p + 7q 10) 5p q pq + 5pq 3p 11q + 18 (4p q + 5pq 3pq 8p + 7q 10) 5p q pq + 5pq 3p 11q + 18 4p q 5pq + 3pq + 8p 7q + 10 p q 7pq + 8pq + 5p 18q + 8 Exercise 9. Q.1. 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 Ans. (i) 4 7p 8p (ii) 4p 7p 8p (iii) 4p 7pq 8p q (iv) 4p 3 ( 3p) 1p 4 (v) 4p 0 0 Q.. Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively. (p, q); (10m, 5n); (0x, 5y ); (4x, 3x ); (3mn, 4np) Ans. (i) Here, l p, b q Area of rectangle p q pq (ii) Here, l 10m, b 5n Area of rectangle 10m 5n 50mn (iii) Here, l 0x, b 5y Area of rectangle 0x 5y 100x y 4

(iv) Here, l 4x, b 3x Area of rectangle 4x 3x 1x 3 (v) Here, l 3mn, b 4np Area of rectangle 3mn 4np Q.3. Complete the table of products. 1mn p Ans. Complete table of products is shown below. Q.4. Obtain the volume of rectangular boxes with the following length, breadth and height respectively. (i) 5a, 3a, 7a 4 (ii) p, 4q, 8r (iii) xy, x y, xy (iv) a, b, 3c 5

Ans. Volume of cuboid l b h (i) l 5a, b 3a, h 7a 4 Volume of rectangular box l b h (ii) l p, b 4q, h 8r 6 5a 3a 7a 4 105a 7 Volume of rectangular box l b h (iii) l xy, b x y, h xy p 4q 8r 64pqr Volume of rectangular box xy x y xy (iv) l a, b b, h 3c 4x 4 y 4 Volume of rectangular box l b h a b 3c 6abc Q.5. Obtain the product of (i) xy, yz, zx (ii) a, a, a 3 (iii), 4y, 8y, 16y 3 (iv) a, b, 3c, 6abc (v) m, mn, mnp Ans. (i) xy yz zx x y z (ii) a ( a ) (a 3 ) a 6 (iii) 4y 8y 16y 3 104y 6 (iv) a b 3c 6abc 36a b c (v) m ( mn) mnp m 3 n p

Exercise 9.3 Q.1. 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 b (iv) a 9, 4a (v) pq + qr + rp, 0 Ans. (i) 4p(q + r) 4pq + 4pr (ii) ab (a b) a b ab (iii) (a + b) (7a b ) 7a 3 b + 7a b 3 (iv) (a 9) 4a 4a 3 36a (v)(pq + qr + rp) 0 0 Q.. Complete the table. Ans. (i) a(b + c + d) ab + ac + ad (ii) (x + y 5)5xy 5x y + 5xy 5xy (iii) p(6p 7p + 5) 6p 3 7p + 5p (iv) (4p q ) (p q ) 4p 4 q 4p q 4 (v) (a + b + c) abc a bc + ab c + abc Q.3.Find the product. (i) (a ) (a ) (4a 6 ) (ii) 9 xy x y 3 10 7

(iii) 10 3 6 pq 3 5 3 p q Ans. (i) (a ) (a ) (4a 6 ) (ii) (iii) 9 xy x 3 10 (iv) x x x 3 x 4 4 a a a 6 8a 50 y 9 3 10 x x y y 3 5 x3 y 3 10 3 6 3 pq p q 3 5 10 6 3 5 p p3 q 3 q 4p 4 q 4 (iv) x x x 3 x 4 x 10 Q.4. (a) Simplify 3x(4x 5) + 3 and find its value for (i) x 3 (ii) x 1. (b) Simplify a(a + a + 1) + 5 and find its value for (i) a 0, (ii) a 1 (iii) a 1. Ans. (a) (i) 3x(4x 5) + 3 1x 15x + 3 ------------ (i) Putting x 3 in (i), we get 1 (3) 15 3 + 3 1 9 45 + 3 108 4 66 Hence, for x 3, 3x (4x 5) + 3 66 8

(ii) Putting x 1 in (i), we get 1 1 15 1 + 3 1 1 1 5 15 + 3 6 4 Hence, for x 1 1 15 3, 3x (4x 5) + 3 3 (b) a(a + a + 1) + 5 a 3 + a + a + 5 (i) For a 0, (ii) For a 1, a 3 + a + a + 5 0 3 + 0 + 0 + 5 5 a 3 + a + a + 5 (1) 3 + 1 + 1 + 5 (iii) For a 1, 1 + 1 + 1 + 5 8 a 3 + a + a + 5 ( 1) 3 + ( 1) + ( 1) + 5 1 + 1 1 + 5 4 Q.5. (a) Add: p(p q), q(q r) and r(r p) (b) Add: x(z x y) and y(z y x) (c) Subtract: 3l(l 4m + 5n) from 4l(10n 3m + l) (d) Subtract: 3a(a + b + c) b(a b + c) from 4c( a + b + c) Ans. (a) (p pq) + (q qr) + (r rp) p + q + r pq qr pr 9

(b) (xz x xy) + (yz y xy) xz x xy + yz y xy x y 4xy + yz + zx (c) (40ln 1lm + 8l ) (3l 1lm + 15ln) 40ln 1lm + 8l 3l + 1lm 15ln 5l + 5ln (d)( 4ca + 4bc + 4c ) (3a + 3ab + 3ac ab + b bc) ( 4ca + 4bc + 4c ) (3a + b + ab bc + 3ac) 4ac + 4bc + 4c 3a b ab + bc 3ac 3a b + 4c ab + 6bc 7ac Exercise 9.4 Q.1. Multiply the binomials. (i) (x + 5) and (4x 3) (ii) (y 8) and (3y 4) (iii) (.5l 0.5m) and (.5l + 0.5m) (iv) (a + 3b) and (x + 5) (v) (pq + 3q ) and (3pq q ) (vi) 3 a + 3 b and 4 a b 4 3 Ans. (i) (x + 5) (4x 3) 8x 6x + 0x 15 8x + 14x 15 (ii) (y 8) (3y 4) 3y 4y 4y + 3 3y 8y + 3 (iii) (.5l 0.5m) (.5l + 0.5m) 6.5l + 1.5lm 1.5lm 0.5m 6.5l 0.5m 10

(iv) (a + 3b) (x + 5) ax + 5a + 3bx + 15b ax + 3bx + 5a + 15b (v)(pq + 3q ) (3pq q ) 6p q 4pq 3 + 9pq 3 6q 4 (vi) 3 4 3 a + 3 b 4 a b 6p q + 5pq 3 6q 4 3 4 a a b + 3 b a b 4 3 3 3 3 b 4 4 3 3 4 4 a a b + 3 a b 3 3 1 a a b a b b 4 4 4 + 3 3 6 ab ab 4 a + b 4 4 4 3 5ab 4 a + b 4 4 4 3a 4 + 8b 4 + 10a b Q.. Find the product. (i) (5 x) (3 + x) (ii) (x + 7y) (7x y) (iii) (a + b) (a + b ) (iv) (p q ) (p + q) Ans. (i) (5 x) (3 + x) 15 + 5x 6x x 15 x x (ii) (x + 7y) (7x y) 7x xy + 49xy 7y 7x 7y + 48xy (iii) (a + b) (a + b ) a 3 + a b + ab + b 3 a 3 + b 3 + a b + ab 4 4 11

(iv)(p q ) (p + q) p 3 + p q pq q 3 p 3 q 3 + p q pq Q.3. Simplify. (i) (x 5) (x + 5) + 5 (ii) (a + 5) (b 3 + 3) + 5 (iii) (t + s ) (t s) (iv) (a + b) (c d) + (a b) (c + d) + (ac + bd) (v) (x + y) (x + y) + (x + y) (x y) (vi) (x + y) (x xy + y ) (vii) (1.5x 4y) (1.5x + 4y + 3) 4.5x + 1y (viii) (a + b + c) (a + b c) Ans. (i) (x 5) (x + 5) + 5 x 3 + 5x 5x 5 + 5 x 3 + 5x 5x (ii) (a + 5) (b 3 + 3) + 5 a b 3 + 3a + 5b 3 + 15 + 5 a b 3 + 3a + 5b 3 + 0 (iii) (t + s ) (t s) t 3 st + s t s 3 t 3 s 3 st + s t (iv) (a + b) (c d) + (a b) (c + d) + (ac + bd) ac ad + bc bd + ac + ad bc bd + ac + bd 4ac (v) (x + y) (x + y) + (x + y) (x y) x + xy + xy + y + x xy + xy y 3x y + 4xy (vi) (x + y) (x xy + y ) x 3 x y + xy + x y xy + y 3 x 3 + y 3 (vii) (1.5x 4y) (1.5x + 4y + 3) 4.5x + 1y.5x + 6xy + 4.5x 6xy 16y 1y 4.5x + 1y.5x 16y 1

(viii) (a + b + c) (a + b c) a + ab ac + ab + b bc + ac + bc c a + b c + ab Exercise 9.5 Q.1. Use a suitable identity to get each of the following products. (i) (x + 3) (x + 3) (ii) (y + 5) (y + 5) (iii) (a 7) (a 7) (iv) (3a 1 ) (3a 1 ) (v) (1.1m 0.4) (1.1m + 0.4) (vi) (a + b ) ( a + b ) (vii) (6x 7) (6x + 7) (viii) ( a + c) ( a + c) (ix) x 3y x 3y + + 4 4 (x) (7a 9b) (7a 9b) Ans. (i)(x + 3) (x + 3) (x + 3) x + 3 x + (3) [ (a + b) a + ab + b ] x + 6x + 9 (ii) (y + 5) (y + 5) (y + 5) (iii)(a 7) (a 7) (a 7) (iv) 1 1 3 a 3 a (y) + y 5 + (5) [ (a + b) a + ab + b ] 4y + 0y + 5 (a) a 7 + (7) [ (a b) a ab + b ] 4a 8a + 49 3 a 1 13

(3a) 3a 1 1 + (v) (1.1m 0.4) (1.1m + 0.4) 9a 3a + 1 4 (1.1m + 0.4) (1.1m 0.4) (1.1m) (0.4) [ (a + b) (a b) a b ] 1.1m 0.16 (vi) (a + b ) ( a + b ) (b + a ) (b a ) (b ) (a ) b 4 a 4 (vii) (6x 7) (6x + 7) (6x) (7) 36x 49 (viii) ( a + c) ( a + c) (c a) (c a) (ix) x 3 3 + y x + y 4 4 (c a) c ca + a x 3y + 4 x 3 3 +. x. y + y 4 4 x 3xy 9y + + 4 4 16 (x) (7a 9b) (7a 9b) (7a 9b) (7a) 7a 9b + (9b) 49a 16ab + 81b 14

Q.. Use the identity (x + a) (x + b) x + (a + b) x + ab 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) (x + 5y) ( x + 3y) (vi) (a + 9) (a + 5) (vii) (xyz 4) (xyz ) Ans. (i) (x + 3) (x + 7) (here a 3, b 7) x + (3 + 7)x + 3 7 x + 10x + 1 (ii) (4x + 5) (4x + 1) (here a 5, b 1) (4x) + (5 + 1) 4x + 5 1 16x + 4x + 5 (iii) (4x 5) (4x 1) (here a 5, b 1) (4x) + ( 5 1)4x + ( 5) ( 1) 16x 4x + 5 (iv) (4x + 5) (4x 1) (here a 5, b 1) (4x) + (5 1)4x + 5 ( 1) 16x + 16x 5 (v) (x + 5y) (x + 3y) (here a 5y, b 3y) (x) + (5y + 3y)x + 5y 3y 4x + 16xy + 15y (vi) (a + 9) (a + 5) (here a 9, b 5, x a ) (a ) + (9 + 5)a + 9 5 4a 4 + 8a + 45 (vii) (xyz 4) (xyz ) (here a 4, b ) (xyz) + ( 4 )xyz + ( 4) ( ) x y z 6xyz + 8 15

Q.3. Find the following squares by using the identities. (i) (b 7) (ii) (xy + 3z) (iii) (6x 5y) 3 (iv) m+ n 3 (v) (0.4p 0.5q) (vi) (xy + 5y) Ans. (i) (b 7) b b 7 + 7 b 14b + 49 [ (a b) a ab + b )] (ii) (xy + 3z) (xy) + xy 3z + (3z) x y + 6xyz + 9z. [ (a + b) a + ab + b ] (iii) (6x 5y) (6x ) 6x 5y + (5y) (iv) 3 m + n 3 36x 4 60x y + 5y 3 m + 3 n + m 3 n 3 4 9 m + n +mn 9 4 (v) (0.4p 0.5q) (0.4p) 0.4p 0.5q + (0.5q) 0.16p 0.40pq + 0.5q (vi) (xy + 5y) (xy) + xy 5y + (5y) 4x y + 0xy + 5y Q.4. Simplify : (i) (a b ) (ii) (x + 5) (x 5) (iii) (7m 8n) + (7m + 8n) (iv) (4m + 5n) + (5m + 4n) (v) (.5p 1.5q) (1.5p.5q) 16

(vi) (ab + bc) ab c (vii) (m n m) + m 3 n Ans. (i) (a b ) (a ) a b + (b ) a 4 a b + b 4 (ii) (x + 5) (x 5) [(x + 5) + (x 5)] [(x + 5) (x 5)] ( x + 5 + x 5 (x + 5 x + 5) 4x 10 40x (iii) (7m 8n) + (7m + 8n) ) (7m) 7m 8n + (8n) + (7m) + 7m 8n + (8n) (7m) + (8n) 49m + 64n 98m + 18n (iv) (4m + 5n) + (5m + 4n) (4m) + 4m 5n + (5n) + (5m) + 5m 4n + (4n) 16m + 40mn + 5n + 5m + 40mn + 16n 41m + 41n + 80mn (v) (.5p 1.5q) (1.5p.5q) [(.5p 1.5q) + (1.5p.5q)] [(.5p 1.5q) (1.5p.5q)] (.5p 1.5q + 1.5p.5q) (.5p 1.5q 1.5p +.5q) (4p 4q) (p + q) 4(p q) (p + q) 4(p q ) 17

(vi) (ab + bc) ab c (ab) + ab bc + (bc) ab c a b + ab c + b c ab c a b + b c (vii) (m n m) + m 3 n (m ) m n m + (n m) + m 3 n m 4 m 3 n + n 4 m + m 3 n m 4 + n 4 m Q.5. Show that. (i) (3x + 7) 84x (3x 7) (ii) (9p 5q) + 180pq (9p + 5q) (iii) 4 3 m n 3 4 + mn 16 9 m + n 9 16 (iv) (4pq + 3q) (4pq 3q) 48pq (v) (a b) (a + b) + (b c) (b + c) + (c a) (c + a) 0 Ans. (i) (3x + 7) 84x (3x 7) L.H.S (3x) + 3x 7 + (7) 84x 9x + 4x + 49 84x 9x 4x + 49 (3x 7) (3x) 3x 7 + (7) R.H.S (ii) (9p 5q) + 180pq (9p + 5q) L.H.S. (9p) 9p 5q + (5q) + 180pq (9p) + 90pq + (5q) (9p) + 9pq 5q + (5q) (9p + 5q) R.H.S. 18

4 3 16 9 (iii) m n + mn m + n 3 4 9 16 L.H.S 4 3 m n 3 4 + mn 4 3 m + n 3 4 4 3 mn + mn 3 4 16 9 m + n mn + mn 9 16 16 9 m + n R.H.S 9 16 (iv) (4pq + 3q) (4pq 3q) 48pq L.H.S. [(4pq + 3q) + (4pq 3q)] [(4pq + 3q) (4pq 3q)] (4pq + 3q + 4pq 3q) (4pq + 3q 4pq + 3q) 8pq 6q 48pq R.H.S (v) (a b) ( a + b) + (b c) (b + c) + (c a) (c + a) 0 L.H.S. a b + b c + c a 0 R.H.S. Q.6. Using identities, evaluate. (i) 71 (ii) 99 (iii) 10 (iv) 998 (v) 5. (vi) 97 303 (vii) 78 8 (viii) 8.9 (ix) 1.05 9.5 Ans. (i) (71) (70 + 1) (70) + 70 1 + (1) 4900 + 140 + 1 5041 19

(ii) (99) (100 1) (100) 100 1 + (1) 10000 00 + 1 9801 (iii) (10) (100 + ) (100) + 100 + () 10000 + 400 + 4 10404 (iv) (998) (1000 ) (1000) 1000 + () 1000000 4000 + 4 996004 (v) (5.) (5 + 0.) (5) + 5 0. + (0.) 5 +.0 + 0.04 7.04 (vi) 97 303 (300 3) (300 + 3) (300) (3) 90000 9 89991 (vii) 78 8 (80 ) (80 + ) (80) () 6400 4 6396 (viii) (8.9) (9 0.1) (9) 9 0.1 + (0.1) 81 1.8 + 0.01 79.1 0

(ix) (1.05) (9.5) ( 1.05 10 ) 9.5 10 10.5 9.5 10 ( 10 + 0.5 ) ( 10 0.5 ) 10 1 ( ) 10 0.5 10 100 0.5 10 99.75 10 9.975 Q.7. Using a b (a + b) (a b), find (i) 51 49 (ii) (1.0) (0.98) (iii) 153 147 (iv) 1.1 7.9 Ans. (i) 51 49 (51 + 49) (51 49) 100 00 (ii) (1.0) (0.98) (1.0 + 0.98) (1.0 0.98).00 0.04 0.08 (iii) (153) (147) (153 + 147) (153 147) 300 6 1800 (iv) (1.1) (7.9) (1.1 + 7.9) (1.1 7.9) 0.0 4. 84.0 84

Q.8. Using (x + a) (x + b) x + (a + b)x + ab, find (i) 103 104 (ii) 5.1 5. (iii) 103 98 (iv) 9.7 9.8 Ans. (i) 103 104 (100 + 3) (100 + 4) (100) + (3 + 4) 100 + 3 4 10000 + 700 + 1 1071 (ii) 5.1 5. (5 + 0.1) (5 + 0.) 5 + (0.1 + 0.) 5 + (0.1 0.) 5 + 1.5 + 0.0 6.5 + 0.0 6.5 (iii) 103 98 (100 + 3) (100 ) (100) + [3 + ( )] 100 + 3 ( ) 10000 + 100 6 10094 (iv) 9.7 9.8 (10 0.3) (10 0.) (10) + ( 0.3 0.) 10 + [( 0.) ( 0.3)] 100 + ( 0.5) 10 + 0.06 100 5 + 0.06 95.06

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