Q1. Multiply the following:
(a) $8ab^2$ by $(-4a^3b^4)$
(b) $\dfrac{2}{3}ab$ by $(-\dfrac{1}{4})a^2b$
(c) $(-5cd^2)$ by $(-5cd^2)$
(d) $4a$ by $6a + 7$
(e) $(-8x)$ and $(4 - 2x - x^2)$
(f) $(2a^2 - 5a - 4)$ and $(-3a)$
(g) $(x + 4)$ by $(x - 5)$
(h) $(5a - 1)$ by $(7a - 3)$
(i) $(12a + 5b)$ by $(7a - b)$
(j) $(x^2 + x +1)$ by $(1 - x)$
(k) $(2m^2 - 3m - 1)$ and $(4m^2 - m - 1)$
(l) $a^2, ab$ and $b^2$
(m) $abx, (-3a^2)$ and $7b^2x^3$
(n) $(-3bx), (-5xy)$ and $(-7b^3y^2)$
(o) $(-\dfrac{3}{2}x^5y^3)$ and $(\dfrac{4}{9}a^2x^3y)$
(p) $(-\dfrac{2}{3}a^7b^2)$ and $(-\dfrac{9}{4}ab^5)$
(q) $2a^3 - 3a^2b$ and $(-\dfrac{1}{2}ab^2)$
(r) $2x + \dfrac{1}{2}y$ and $2x - \dfrac{1}{2}y$
Show Answer
(a) $8ab^2 \times (-4a^3b^4)$
$\Rightarrow ((8 \times (-4))a^{1+3}b^{2+4}) = (-32a^4b^6)$ Ans
(b) $\dfrac{2}{3}ab$ by $(-\dfrac{1}{4})a^2b$
$\Rightarrow (-\dfrac{2 \times 1}{3 \times 4})a^{1+2}b^{1+1} = (-\dfrac{2}{12})a^3b^2 = (-\dfrac{1}{6})a^3b^2$ Ans
(c) $(-5cd^2) \text{ by } (-5cd^2)$
$\Rightarrow (-5) \times (-5)c^{1+1}d^{2+2} = 25c^2d^4$ Ans
(d) $4a \text{ and } 6a + 7$
$\Rightarrow 4a(6a + 7) = (4 \times 6)a^{1+1} + (4 \times 7)a = 24a^2 + 28a$ Ans
(e) $(-8x) \text{ and } 4 - 2x + x^2$
$\Rightarrow ((-8) \times 4)x + ((-8) \times (-2))x^{1+1} + ((-8) \times 1)x^{1+2}$
$\Rightarrow (-32x) + 16x^2 + (-8x^3) = (-32x) + 16x^2 - 8x^3$ Ans
(f) $(2a^2 - 5a - 4)$ and $(-3a)$
$\Rightarrow ((2 \times (-3))a^{2+1} + ((-5) \times (-3))a^{1+1} + ((-4) \times (-3))a$
$\Rightarrow (-6)a^3 + 15a^2 + 12a$ Ans
(g) $(x + 4)$ by $(x - 5)$
$\Rightarrow x(x - 5) + 4(x - 5)$
$\Rightarrow x^2 - 5x + 4x - 20$
$\Rightarrow x^2 - x - 20$ Ans
(h) $(5a - 1)$ by $(7a - 3)$
$\Rightarrow 5a(7a - 3) - 1(7a - 3)$
$\Rightarrow 35a^{1+1} - 15a - 7a + 3$
$\Rightarrow 35a^2 - 22a + 3$ Ans
(i) $(12a + 5b)$ by $(7a - b)$
$\Rightarrow 12a(7a - b) + 5b(7a - b)$
$\Rightarrow 84a^{1+1} - 12ab + 35ab - 5b^{1+1}$
$\Rightarrow 84a^2 + 23ab - 5b^2$ Ans
(j) $(x^2 + x +1)$ by $(1 - x)$
$\Rightarrow 1(x^2 + x + 1) - x(x^2 + x +1)$
$\Rightarrow x^2 + x + 1 - x^{1+2} - x^{1+1} - x$
$\Rightarrow x^2 + x + 1 - x^3 - x^2 - x$
$\Rightarrow (x^2 - x^2) + (x- x) + 1 - x^3$
$\Rightarrow 1 - x^3$ Ans
(k) $(2m^2 - 3m - 1)$ and $(4m^2 - m - 1)$
$\Rightarrow 2m^2(4m^2 - m - 1) - 3m(4m^2 - m - 1) - 1(4m^2 - m - 1)$
$\Rightarrow 8m^{2+2} - 2m^{2+1} - 2m^2 - 12m^{1+2} + 3m^{1+1} + 3m - 4m^2 + m + 1$
$\Rightarrow 8m^4 - 2m^3 - 2m^2 - 12m^3 + 3m^2 + 3m - 4m^2 + m + 1$
$\Rightarrow 8m^4 + (-2m^3 - 12m^3) + (-2m^2 + 3m^2 - 4m^2) + (3m + m) + 1$
$\Rightarrow 8m^4 + (-14m^3) + (-3m^2) + (4m) + 1$
$\Rightarrow 8m^4 - 14m^3 - 3m^2 + 4m + 1$ Ans
(l) $a^2, ab$ and $b^2$
$\Rightarrow b^2(a^2 \times ab)$
$\Rightarrow b^2(a^{1+2}b)$
$\Rightarrow a^{1+2}b^{2+1}$
$\Rightarrow a^3b^3$ Ans
(m) $abx, (-3a^2)$ and $7b^2x^3$
$\Rightarrow abx((-3 \times 7)a^2b^2x^3)$
$\Rightarrow (-21a^{2+1}b^{2+1}x^{3+1})$
$\Rightarrow -21a^3b^3x^4$ Ans
(n) $(-3bx), (-5xy)$ and $(-7b^3y^2)$
$\Rightarrow -3bx ((-5) \times (-7))b^3xy^{1+2}$
$\Rightarrow -3bx (35b^3xy^3)$
$\Rightarrow (-3) \times 35b^{3+1}x^{1+1}y^3$
$\Rightarrow -105b^4x^2y^3$ Ans
(o) $(-\dfrac{3}{2}x^5y^3)$ and $(\dfrac{4}{9}a^2x^3y)$
$\Rightarrow ((-\dfrac{3}{2}) \times \dfrac{4}{9})a^2x^{5+3}y^{3+1}$
$\Rightarrow (-\dfrac{12}{18})a^2x^8y^4 = (-\dfrac{2}{3})a^2x^8y^4$ Ans
(p) $(-\dfrac{2}{3}a^7b^2)$ and $(-\dfrac{9}{4}ab^5$
$\Rightarrow (-\dfrac{2}{3}) \times (-\dfrac{9}{4})a^{7+1}b^{2+5}$
$\Rightarrow \dfrac{18}{12}a^8b^7 = \dfrac{3}{2}a^8b^7$ Ans
(q) $2a^3 - 3a^2b$ and $(-\dfrac{1}{2}ab^2)$
$\Rightarrow ((-\dfrac{1}{2}) \times 2)a^{3+1}b^2 - ((-\dfrac{1}{2}) \times 3)a^{2+1}b^{1+2}$
$\Rightarrow -a^4b^2 + \dfrac{3}{2}a^3b^3$ Ans
(r) $2x + \dfrac{1}{2}y$ and $2x - \dfrac{1}{2}y$
$\Rightarrow 2x(2x) - \dfrac{1}{2}y(\dfrac{1}{2}y)$
$\Rightarrow 4x^{1+1} - \dfrac{}{4}y^{1+1}$
$\Rightarrow 4x^2 - \dfrac{1}{4}y^2$ Ans
$\Rightarrow ((8 \times (-4))a^{1+3}b^{2+4}) = (-32a^4b^6)$ Ans
(b) $\dfrac{2}{3}ab$ by $(-\dfrac{1}{4})a^2b$
$\Rightarrow (-\dfrac{2 \times 1}{3 \times 4})a^{1+2}b^{1+1} = (-\dfrac{2}{12})a^3b^2 = (-\dfrac{1}{6})a^3b^2$ Ans
(c) $(-5cd^2) \text{ by } (-5cd^2)$
$\Rightarrow (-5) \times (-5)c^{1+1}d^{2+2} = 25c^2d^4$ Ans
(d) $4a \text{ and } 6a + 7$
$\Rightarrow 4a(6a + 7) = (4 \times 6)a^{1+1} + (4 \times 7)a = 24a^2 + 28a$ Ans
(e) $(-8x) \text{ and } 4 - 2x + x^2$
$\Rightarrow ((-8) \times 4)x + ((-8) \times (-2))x^{1+1} + ((-8) \times 1)x^{1+2}$
$\Rightarrow (-32x) + 16x^2 + (-8x^3) = (-32x) + 16x^2 - 8x^3$ Ans
(f) $(2a^2 - 5a - 4)$ and $(-3a)$
$\Rightarrow ((2 \times (-3))a^{2+1} + ((-5) \times (-3))a^{1+1} + ((-4) \times (-3))a$
$\Rightarrow (-6)a^3 + 15a^2 + 12a$ Ans
(g) $(x + 4)$ by $(x - 5)$
$\Rightarrow x(x - 5) + 4(x - 5)$
$\Rightarrow x^2 - 5x + 4x - 20$
$\Rightarrow x^2 - x - 20$ Ans
(h) $(5a - 1)$ by $(7a - 3)$
$\Rightarrow 5a(7a - 3) - 1(7a - 3)$
$\Rightarrow 35a^{1+1} - 15a - 7a + 3$
$\Rightarrow 35a^2 - 22a + 3$ Ans
(i) $(12a + 5b)$ by $(7a - b)$
$\Rightarrow 12a(7a - b) + 5b(7a - b)$
$\Rightarrow 84a^{1+1} - 12ab + 35ab - 5b^{1+1}$
$\Rightarrow 84a^2 + 23ab - 5b^2$ Ans
(j) $(x^2 + x +1)$ by $(1 - x)$
$\Rightarrow 1(x^2 + x + 1) - x(x^2 + x +1)$
$\Rightarrow x^2 + x + 1 - x^{1+2} - x^{1+1} - x$
$\Rightarrow x^2 + x + 1 - x^3 - x^2 - x$
$\Rightarrow (x^2 - x^2) + (x- x) + 1 - x^3$
$\Rightarrow 1 - x^3$ Ans
(k) $(2m^2 - 3m - 1)$ and $(4m^2 - m - 1)$
$\Rightarrow 2m^2(4m^2 - m - 1) - 3m(4m^2 - m - 1) - 1(4m^2 - m - 1)$
$\Rightarrow 8m^{2+2} - 2m^{2+1} - 2m^2 - 12m^{1+2} + 3m^{1+1} + 3m - 4m^2 + m + 1$
$\Rightarrow 8m^4 - 2m^3 - 2m^2 - 12m^3 + 3m^2 + 3m - 4m^2 + m + 1$
$\Rightarrow 8m^4 + (-2m^3 - 12m^3) + (-2m^2 + 3m^2 - 4m^2) + (3m + m) + 1$
$\Rightarrow 8m^4 + (-14m^3) + (-3m^2) + (4m) + 1$
$\Rightarrow 8m^4 - 14m^3 - 3m^2 + 4m + 1$ Ans
(l) $a^2, ab$ and $b^2$
$\Rightarrow b^2(a^2 \times ab)$
$\Rightarrow b^2(a^{1+2}b)$
$\Rightarrow a^{1+2}b^{2+1}$
$\Rightarrow a^3b^3$ Ans
(m) $abx, (-3a^2)$ and $7b^2x^3$
$\Rightarrow abx((-3 \times 7)a^2b^2x^3)$
$\Rightarrow (-21a^{2+1}b^{2+1}x^{3+1})$
$\Rightarrow -21a^3b^3x^4$ Ans
(n) $(-3bx), (-5xy)$ and $(-7b^3y^2)$
$\Rightarrow -3bx ((-5) \times (-7))b^3xy^{1+2}$
$\Rightarrow -3bx (35b^3xy^3)$
$\Rightarrow (-3) \times 35b^{3+1}x^{1+1}y^3$
$\Rightarrow -105b^4x^2y^3$ Ans
(o) $(-\dfrac{3}{2}x^5y^3)$ and $(\dfrac{4}{9}a^2x^3y)$
$\Rightarrow ((-\dfrac{3}{2}) \times \dfrac{4}{9})a^2x^{5+3}y^{3+1}$
$\Rightarrow (-\dfrac{12}{18})a^2x^8y^4 = (-\dfrac{2}{3})a^2x^8y^4$ Ans
(p) $(-\dfrac{2}{3}a^7b^2)$ and $(-\dfrac{9}{4}ab^5$
$\Rightarrow (-\dfrac{2}{3}) \times (-\dfrac{9}{4})a^{7+1}b^{2+5}$
$\Rightarrow \dfrac{18}{12}a^8b^7 = \dfrac{3}{2}a^8b^7$ Ans
(q) $2a^3 - 3a^2b$ and $(-\dfrac{1}{2}ab^2)$
$\Rightarrow ((-\dfrac{1}{2}) \times 2)a^{3+1}b^2 - ((-\dfrac{1}{2}) \times 3)a^{2+1}b^{1+2}$
$\Rightarrow -a^4b^2 + \dfrac{3}{2}a^3b^3$ Ans
(r) $2x + \dfrac{1}{2}y$ and $2x - \dfrac{1}{2}y$
$\Rightarrow 2x(2x) - \dfrac{1}{2}y(\dfrac{1}{2}y)$
$\Rightarrow 4x^{1+1} - \dfrac{}{4}y^{1+1}$
$\Rightarrow 4x^2 - \dfrac{1}{4}y^2$ Ans
Q2. Multiply:
(a) $5x^2 - 8xy + 6y^2 - 3$ by $(-3xy)$
(b) $3 - \dfrac{2}{3}xy + \dfrac{5}{7}xy^2 - \dfrac{16}{21}x^2y$ by $(-21x^2y^2)$
(c) $6x^3 - 5x + 10$ by $4 - 3x^2$
(d) $2y - 4y^3 + 6y^5$ by $y^2 + y - 3$
(e) $5p^2 + 25pq + 4q^2$ by $2p^2 - 2pq + 3q^2$
Show Answer
(a) $5x^2 - 8xy + 6y^2 - 3$ by $(-3xy)$
$\Rightarrow -3xy(5x^2 - 8xy + 6y^2 - 3)$
$\Rightarrow (-3)5x^{2+1}y - (-3)8x^{1+1}y^{1+1} + (-3)6xy^{2+1} - (-3)3xy$
$\Rightarrow -15x^3y - (-24)x^2y^2 + (-18)xy^3 - (-9)xy$
$\Rightarrow -15x^3y + 24x^2y^2 - 18xy^3 + 9xy$ Ans
(b) $3 - \dfrac{2}{3}xy + \dfrac{5}{7}xy^2 - \dfrac{16}{21}x^2y$ by $(-21x^2y^2)$
$\Rightarrow -21x^2y^2(3 - \dfrac{2}{3}xy + \dfrac{5}{7}xy^2 - \dfrac{16}{21}x^2y$
$\Rightarrow {(-21)3x^2y^2 - (-21) \times \dfrac{2}{3}x^{2+1}y^{2+1} + (-21) \times (-21) \times \dfrac{5}{7}x^{2+1}y^{2+2} - (-21) \times \dfrac{16}{21}x^{2+1}y^2}$
$\Rightarrow (-63)x^2y^2 - (-7) \times 2x^3y^3 + (-3) \times 5x^3y^4 - (-16)x^3y^2$
$\Rightarrow (-63)x^2y^2 - (-14)x^3y^3 + (-15)x^3y^4 - (-16)x^3y^2$
$\Rightarrow -63x^2y^2 - 14x^3y^3 - 15x^3y^4 + 16x^3y^2$ Ans
(c) $6x^3 - 5x + 10$ by $4 - 3x^2$
$\Rightarrow 4 - 3x^2(6x^3 - 5x + 10)$
$\Rightarrow 4(6x^3 - 5x + 10) - 3x^2(6x^3 - 5x + 10)$
$\Rightarrow (4)6x^3 - (4)5x + (4)10 - (3)6x^{3+2} - (-3)5x^{2+1} + (-3)10x^2)$
$\Rightarrow 24x^3 - 20x + 40 - 18x^5 - (-15)x^3 + (-30)x^2$
$\Rightarrow 24x^3 - 20x + 40 - 18x^5 + 15x^3 - 30x^2$
$\Rightarrow - 18x^5 + (24 + 15)x^3 - 30x^2 - 20x + 40$
$\Rightarrow - 18x^5 + 39x^3 - 30x^2 - 20x + 40$ Ans
(d) $2y - 4y^3 + 6y^5$ by $y^2 + y - 3$
$\Rightarrow y^2 + y - 3(2y - 4y^3 + 6y^5)$
$\Rightarrow y^2(2y - 4y^3 + 6y^5) + y(2y - 4y^3 + 6y^5) - 3(2y - 4y^3 + 6y^5)$
$\Rightarrow {(2y^{2+1} - 4y^{3+2} + 6y^{5+2}) + (2y^{1+1} - 4y^{3+1} + 6y^{5+1}) - (3)2y - (-3)4y^3 + (-3)6y^5}$
$\Rightarrow 2y^3 - 4y^5 + 6y^7 + 2y^2 - 4y^4 + 6y^6 - 6y - (-12)y^3 + (-18)y^5$
$\Rightarrow 2y^3 - 4y^5 + 6y^7 + 2y^2 - 4y^4 + 6y^6 - 6y + 12y^3 - 18y^5$
$\Rightarrow 6y^7 + 6y^6 + (-4 - 18)y^5 - 4y^4 + (12 + 2)y^3 + 2y^2 - 6y$
$\Rightarrow 6y^7 + 6y^6 + (-22)y^5 - 4y^4 + 14y^3 + 2y^2 - 6y$
$\Rightarrow 6y^7 + 6y^6 - 22y^5 - 4y^4 + 14y^3 + 2y^2 - 6y$ Ans
(e) $5p^2 + 25pq + 4q^2$ by $2p^2 - 2pq + 3q^2$
$\Rightarrow 2p^2 - 2pq + 3q^2(5p^2 + 25pq + 4q^2)$
$\Rightarrow 2p^2(5p^2 + 25pq + 4q^2) - 2pq(5p^2 + 25pq + 4q^2) + 3q^2(5p^2 + 25pq + 4q^2)$
$\Rightarrow {((2)5p^{2+2} + (2)25p^{2+1}q + (2)4p^2q^2) - ((2)5p^{2+1}q + (-2)25p^{1+1}q^{1+1} + (-2)4pq^{2+1}) + ((3)5p^2q^2 + (3)25pq^{2+1} + (3)4q^{2+2})}$
$\Rightarrow 10p^4 + 50p^3q + 8p^2q^2 - 10p^3q - 50p^2q^2 - 8pq^3 + 15p^2q^2 + 75pq^3 + 12q^4$
$\Rightarrow 10p^4 + (50 - 10)p^3q + (8 - 50 + 15)p^2q^2 + (75 - 8)pq^3 + 12q^4$
$\Rightarrow 10p^4 + 40p^3q + (23 - 50)p^2q^2 + 67pq^3 + 12q^4$
$\Rightarrow 10p^4 + 40p^3q + (-27)p^2q^2 + 67pq^3 + 12q^4$
$\Rightarrow 10p^4 + 40p^3q - 27p^2q^2 + 67pq^3 + 12q^4$ Ans
$\Rightarrow -3xy(5x^2 - 8xy + 6y^2 - 3)$
$\Rightarrow (-3)5x^{2+1}y - (-3)8x^{1+1}y^{1+1} + (-3)6xy^{2+1} - (-3)3xy$
$\Rightarrow -15x^3y - (-24)x^2y^2 + (-18)xy^3 - (-9)xy$
$\Rightarrow -15x^3y + 24x^2y^2 - 18xy^3 + 9xy$ Ans
(b) $3 - \dfrac{2}{3}xy + \dfrac{5}{7}xy^2 - \dfrac{16}{21}x^2y$ by $(-21x^2y^2)$
$\Rightarrow -21x^2y^2(3 - \dfrac{2}{3}xy + \dfrac{5}{7}xy^2 - \dfrac{16}{21}x^2y$
$\Rightarrow {(-21)3x^2y^2 - (-21) \times \dfrac{2}{3}x^{2+1}y^{2+1} + (-21) \times (-21) \times \dfrac{5}{7}x^{2+1}y^{2+2} - (-21) \times \dfrac{16}{21}x^{2+1}y^2}$
$\Rightarrow (-63)x^2y^2 - (-7) \times 2x^3y^3 + (-3) \times 5x^3y^4 - (-16)x^3y^2$
$\Rightarrow (-63)x^2y^2 - (-14)x^3y^3 + (-15)x^3y^4 - (-16)x^3y^2$
$\Rightarrow -63x^2y^2 - 14x^3y^3 - 15x^3y^4 + 16x^3y^2$ Ans
(c) $6x^3 - 5x + 10$ by $4 - 3x^2$
$\Rightarrow 4 - 3x^2(6x^3 - 5x + 10)$
$\Rightarrow 4(6x^3 - 5x + 10) - 3x^2(6x^3 - 5x + 10)$
$\Rightarrow (4)6x^3 - (4)5x + (4)10 - (3)6x^{3+2} - (-3)5x^{2+1} + (-3)10x^2)$
$\Rightarrow 24x^3 - 20x + 40 - 18x^5 - (-15)x^3 + (-30)x^2$
$\Rightarrow 24x^3 - 20x + 40 - 18x^5 + 15x^3 - 30x^2$
$\Rightarrow - 18x^5 + (24 + 15)x^3 - 30x^2 - 20x + 40$
$\Rightarrow - 18x^5 + 39x^3 - 30x^2 - 20x + 40$ Ans
(d) $2y - 4y^3 + 6y^5$ by $y^2 + y - 3$
$\Rightarrow y^2 + y - 3(2y - 4y^3 + 6y^5)$
$\Rightarrow y^2(2y - 4y^3 + 6y^5) + y(2y - 4y^3 + 6y^5) - 3(2y - 4y^3 + 6y^5)$
$\Rightarrow {(2y^{2+1} - 4y^{3+2} + 6y^{5+2}) + (2y^{1+1} - 4y^{3+1} + 6y^{5+1}) - (3)2y - (-3)4y^3 + (-3)6y^5}$
$\Rightarrow 2y^3 - 4y^5 + 6y^7 + 2y^2 - 4y^4 + 6y^6 - 6y - (-12)y^3 + (-18)y^5$
$\Rightarrow 2y^3 - 4y^5 + 6y^7 + 2y^2 - 4y^4 + 6y^6 - 6y + 12y^3 - 18y^5$
$\Rightarrow 6y^7 + 6y^6 + (-4 - 18)y^5 - 4y^4 + (12 + 2)y^3 + 2y^2 - 6y$
$\Rightarrow 6y^7 + 6y^6 + (-22)y^5 - 4y^4 + 14y^3 + 2y^2 - 6y$
$\Rightarrow 6y^7 + 6y^6 - 22y^5 - 4y^4 + 14y^3 + 2y^2 - 6y$ Ans
(e) $5p^2 + 25pq + 4q^2$ by $2p^2 - 2pq + 3q^2$
$\Rightarrow 2p^2 - 2pq + 3q^2(5p^2 + 25pq + 4q^2)$
$\Rightarrow 2p^2(5p^2 + 25pq + 4q^2) - 2pq(5p^2 + 25pq + 4q^2) + 3q^2(5p^2 + 25pq + 4q^2)$
$\Rightarrow {((2)5p^{2+2} + (2)25p^{2+1}q + (2)4p^2q^2) - ((2)5p^{2+1}q + (-2)25p^{1+1}q^{1+1} + (-2)4pq^{2+1}) + ((3)5p^2q^2 + (3)25pq^{2+1} + (3)4q^{2+2})}$
$\Rightarrow 10p^4 + 50p^3q + 8p^2q^2 - 10p^3q - 50p^2q^2 - 8pq^3 + 15p^2q^2 + 75pq^3 + 12q^4$
$\Rightarrow 10p^4 + (50 - 10)p^3q + (8 - 50 + 15)p^2q^2 + (75 - 8)pq^3 + 12q^4$
$\Rightarrow 10p^4 + 40p^3q + (23 - 50)p^2q^2 + 67pq^3 + 12q^4$
$\Rightarrow 10p^4 + 40p^3q + (-27)p^2q^2 + 67pq^3 + 12q^4$
$\Rightarrow 10p^4 + 40p^3q - 27p^2q^2 + 67pq^3 + 12q^4$ Ans
Q3. Simplify:
(a) $(7x - 8) (3x + 2)$
(b) $(px - q) (px + q)$
(c) $(5a + 5b - c) (2b - 3c)$
(d) $(4x - 5y) (5x - 4y)$
(e) $(3y + 4z) (3y - 4z) + (2y + 7z) (y + z)$
Show Answer
(a) $(7x - 8) (3x + 2)$
$\Rightarrow 7x(3x + 2) - 8(3x + 2)$
$\Rightarrow 21x^{1+1} + 14x - 24x - 16$
$\Rightarrow 21x^2 + (14 - 24)x - 16$
$\Rightarrow 21x^2 + (-10)x - 16$
$\Rightarrow 21x^2 - 10x - 16$ Ans
(b) $(px - q) (px + q)$
$\Rightarrow px(px + q) - q(px + q)$
$\Rightarrow p^{1+1}x^{1+1} + pxq - pxq - q^{1+1}$
$\Rightarrow p^2x^2 + pxq - pxq - q^2$
$\Rightarrow p^2x^2 - q^2$ Ans
(c) $(5a + 5b - c) (2b - 3c)$
$\Rightarrow 2b(5a + 5b - c) - 3c(5a + 5b - c)$
$\Rightarrow 10ab + 10b^{1+1} - 2bc - 15ac - 15bc + 3c^{1+1}$
$\Rightarrow 10ab + 10b^2 + (-2 - 15)bc - 15ac + 3c^2$
$\Rightarrow 10ab + 10b^2 + (-17)bc - 15ac + 3c^2$
$\Rightarrow 10ab + 10b^2 - 17bc - 15ac + 3c^2$ Ans
(d) $(4x - 5y) (5x - 4y)$
$\Rightarrow 4x(5x - 4y) - 5y(5x - 4y)$
$\Rightarrow 20x^{1+1} - 16xy - 25xy - 20y^{1+1}$
$\Rightarrow 20x^2 + (-16 - 25)xy - 20y^2$
$\Rightarrow 20x^2 + (-41)xy - 20y^2$
$\Rightarrow 20x^2 - 41xy - 20y^2$ Ans
(e) $(3y + 4z) (3y - 4z) + (2y + 7z) (y + z)$
$\Rightarrow 3y(3y - 4z) + 4z(3y - 4z) + 2y(y + z) + 7z(y + z)$
$\Rightarrow 9y^{1+1} - 12yz + 12yz - 16z^{1+1} + 2y^{1+1} + 2yz + 7yz + 7z^{1+1}$
$\Rightarrow 9y^2 - 12yz + 12yz - 16z^2 + 2y^2 + 2yz + 7yz + 7z^2$
$\Rightarrow (9 + 2)y^2 + (-12 + 12 + 2 + 7)yz + (-16 + 7)z^2$
$\Rightarrow 11y^2 + (2 + 7)yz + (-9)z^2$
$\Rightarrow 11y^2 + 9yz - 9z^2$ Ans
$\Rightarrow 7x(3x + 2) - 8(3x + 2)$
$\Rightarrow 21x^{1+1} + 14x - 24x - 16$
$\Rightarrow 21x^2 + (14 - 24)x - 16$
$\Rightarrow 21x^2 + (-10)x - 16$
$\Rightarrow 21x^2 - 10x - 16$ Ans
(b) $(px - q) (px + q)$
$\Rightarrow px(px + q) - q(px + q)$
$\Rightarrow p^{1+1}x^{1+1} + pxq - pxq - q^{1+1}$
$\Rightarrow p^2x^2 + pxq - pxq - q^2$
$\Rightarrow p^2x^2 - q^2$ Ans
(c) $(5a + 5b - c) (2b - 3c)$
$\Rightarrow 2b(5a + 5b - c) - 3c(5a + 5b - c)$
$\Rightarrow 10ab + 10b^{1+1} - 2bc - 15ac - 15bc + 3c^{1+1}$
$\Rightarrow 10ab + 10b^2 + (-2 - 15)bc - 15ac + 3c^2$
$\Rightarrow 10ab + 10b^2 + (-17)bc - 15ac + 3c^2$
$\Rightarrow 10ab + 10b^2 - 17bc - 15ac + 3c^2$ Ans
(d) $(4x - 5y) (5x - 4y)$
$\Rightarrow 4x(5x - 4y) - 5y(5x - 4y)$
$\Rightarrow 20x^{1+1} - 16xy - 25xy - 20y^{1+1}$
$\Rightarrow 20x^2 + (-16 - 25)xy - 20y^2$
$\Rightarrow 20x^2 + (-41)xy - 20y^2$
$\Rightarrow 20x^2 - 41xy - 20y^2$ Ans
(e) $(3y + 4z) (3y - 4z) + (2y + 7z) (y + z)$
$\Rightarrow 3y(3y - 4z) + 4z(3y - 4z) + 2y(y + z) + 7z(y + z)$
$\Rightarrow 9y^{1+1} - 12yz + 12yz - 16z^{1+1} + 2y^{1+1} + 2yz + 7yz + 7z^{1+1}$
$\Rightarrow 9y^2 - 12yz + 12yz - 16z^2 + 2y^2 + 2yz + 7yz + 7z^2$
$\Rightarrow (9 + 2)y^2 + (-12 + 12 + 2 + 7)yz + (-16 + 7)z^2$
$\Rightarrow 11y^2 + (2 + 7)yz + (-9)z^2$
$\Rightarrow 11y^2 + 9yz - 9z^2$ Ans
Q4. The adjacent sides of a rectangle are $x^2 - 4xy + 7y^2$ and $x^3 - 5xy^2$. Find its area.
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$\because$ Area of rectangle = Product of adjacent sides
$\Rightarrow$ Let lenght = $x^2 - 4xy + 7y^2$
$\Rightarrow$ Let breadth = $x^3 - 5xy^2$
$\therefore$ Required area = $(x^3 - 5xy^2) (x^2 - 4xy + 7y^2)$
$\Rightarrow x^3(x^2 - 4xy + 7y^2) - 5xy^2(x^2 - 4xy + 7y^2)$
$\Rightarrow x^{3+2} - 4x^{3+1}y + 7x^3y^2 - 5x^{1+2}y^2 - (-20)x^{1+1}y^{2+1} + (-35)xy^{2+2}$
$\Rightarrow x^5 - 4x^4y + 7x^3y^2 - 5x^3y^2 + 20x^2y^3 - 35xy^4$
$\Rightarrow x^5 - 4x^4y + (7 - 5)x^3y^2 + 20x^2y^3 - 35xy^4$
$\Rightarrow x^5 - 4x^4y + 2x^3y^2 + 20x^2y^3 - 35xy^4$ Ans
$\Rightarrow$ Let lenght = $x^2 - 4xy + 7y^2$
$\Rightarrow$ Let breadth = $x^3 - 5xy^2$
$\therefore$ Required area = $(x^3 - 5xy^2) (x^2 - 4xy + 7y^2)$
$\Rightarrow x^3(x^2 - 4xy + 7y^2) - 5xy^2(x^2 - 4xy + 7y^2)$
$\Rightarrow x^{3+2} - 4x^{3+1}y + 7x^3y^2 - 5x^{1+2}y^2 - (-20)x^{1+1}y^{2+1} + (-35)xy^{2+2}$
$\Rightarrow x^5 - 4x^4y + 7x^3y^2 - 5x^3y^2 + 20x^2y^3 - 35xy^4$
$\Rightarrow x^5 - 4x^4y + (7 - 5)x^3y^2 + 20x^2y^3 - 35xy^4$
$\Rightarrow x^5 - 4x^4y + 2x^3y^2 + 20x^2y^3 - 35xy^4$ Ans
Q5. The base and altitude of a triangle are $3x - 4y$ and $6x + 5y$ respectively. Find its area.
Show Answer
$\because$ Area of a triangle = $\dfrac{1}{2}(\text{lenght } \times \text{ breadth})$
$\Rightarrow$ Let lenght = $3x - 4y$
$\Rightarrow$ Let breadth = $6x + 5y$
$\therefore$ Required area = $\dfrac{1}{2}[(3x - 4y) (6x + 5y)]$
$\Rightarrow \dfrac{1}{2}[3x(6x + 5y) - 4y(6x + 5y)]$
$\Rightarrow \dfrac{1}{2}[18x^{1+1} + 15xy - 24xy + (-20)y^{1+1}]$
$\Rightarrow \dfrac{1}{2}[18x^2 + (15 - 24)xy - 20y^2]$
$\Rightarrow \dfrac{1}{2}[18x^2 + (-9)xy - 20y^2]$
$\Rightarrow \dfrac{1}{2}(18x^2 - 9xy - 20y^2)$ Ans
$\Rightarrow$ Let lenght = $3x - 4y$
$\Rightarrow$ Let breadth = $6x + 5y$
$\therefore$ Required area = $\dfrac{1}{2}[(3x - 4y) (6x + 5y)]$
$\Rightarrow \dfrac{1}{2}[3x(6x + 5y) - 4y(6x + 5y)]$
$\Rightarrow \dfrac{1}{2}[18x^{1+1} + 15xy - 24xy + (-20)y^{1+1}]$
$\Rightarrow \dfrac{1}{2}[18x^2 + (15 - 24)xy - 20y^2]$
$\Rightarrow \dfrac{1}{2}[18x^2 + (-9)xy - 20y^2]$
$\Rightarrow \dfrac{1}{2}(18x^2 - 9xy - 20y^2)$ Ans