સ્વાધ્યાય: અવયવીકરણ

(i) $$12x, 36$$ $$12x$$ ના અવયવ $$= 2\times 2\times 3\times x$$ $$36$$ ના અવયવ $$= 2\times 2\times 3\times 3$$ $$12x$$ અને $$36$$ ના સામાન્ય અવયવ $$= 2\times 2 \times 3 = 12$$ --- (ii) $$2y, 22xy$$ $$2y$$ ના અવયવ $$= 2\times y$$ $$22xy$$ ના અવયવ $$= 2\times 11\times x\times y$$ $$12x$$ અને $$36$$ ના સામાન્ય અવયવ $$= 2\times y = 2y$$ --- (iii) $$14pq, 28p^2q^2$$ $$14pq$$ ના અવયવ $$= 2\times 7\times p\times q$$ $$28p^2q^2$$ ના અવયવ $$= 2\times 2\times 7\times p\times p\times q\times q$$ $$12x$$ અને $$36$$ ના સામાન્ય અવયવ $$= 2\times 7\times p\times q = 14pq$$ --- (iv) $$2x, 3x^2, 4$$ $$2x$$ ના અવયવ $$= 2\times x$$ $$3x^2$$ ના અવયવ $$= 2\times x\times x$$ $$4 $$ના અવયવ $$= 2\times 2$$ $$2x, 3x^2$$ અને $$4$$ ના સામાન્ય અવયવ $$= 1$$ --- (v) $$6abc, 24ab^2, 12a^2b$$ $$6abc$$ ના અવયવ $$= 2\times 3\times a\times b\times c$$ $$24ab^2$$ ના અવયવ $$= 2\times 2\times 2\times 3\times a\times b\times b$$ $$12a^2b$$ ના અવયવ $$= 2\times 2\times 3\times a\times a\times b$$ $$6abc, 24ab^2$$ અને $$12a^2b$$ ના સામાન્ય અવયવ $$= 2\times 3\times a\times b = 6ab$$ --- (vi) $$16x^3, -4x^2, 32x$$ $$16x^3$$ ના અવયવ $$= 2\times 2\times 2\times 2\times x\times x\times x$$ $$-4x^2$$ ના અવયવ $$= -1\times 2\times 2\times x\times x$$ $$32x$$ ના અવયવ $$= 2\times 2 \times 2\times 2\times 2\times x$$ $$16x^3, -4x^2$$ અને $$32x$$ ના સામાન્ય અવયવ $$= 2\times 2\times x = 4x$$ --- (vii) $$10pq, 20gr, 30rp$$ $$10pq$$ ના અવયવ $$= 2\times 5\times p\times q$$ $$20gr$$ ના અવયવ $$= 2\times 2\times 5\times q\times r$$ $$30rp$$ ના અવયવ $$= 2\times 3\times 5\times r\times p$$ $$10pq, 20gr$$ અને $$30rp$$ ના સામાન્ય અવયવ $$= 2\times 5 = 10$$ --- (viii) $$3x2y^3, 10x^3y^2, 6x^2y^2z$$ $$3x2y^3$$ ના અવયવ $$= 3\times x\times x\times y\times y\times y$$ $$10x^3y^2$$ ના અવયવ $$= 2\times 5\times x\times x\times x\times y\times y$$ $$6x^2y^2z$$ ના અવયવ $$= 3\times 2\times x\times x\times y\times y\times z$$ $$16x^3, -4x^2$$ અને $$32x$$ ના સામાન્ય અવયવ $$= x^2\times y^2 = x^2y^2$$

(i) $$7x - 42$$ $$= (7\times x)-(2\times 3\times 7)$$ $$=7(x-6)$$ --- (ii) $$6p - 12q$$ $$= (2\times 3\times p)-(2\times 2\times 3\times q)$$ $$= 2\times 3[p-(2\times q)]$$ $$= 6(p-2q)$$ --- (iii) $$7a^2 + 14a$$ $$= (7 \times a\times a) + (2\times 7\times a)$$ $$= 7 \times a [a + 2] $$ $$= 7a (a + 2)$$ --- (iv) $$-16z + 20z^3$$ $$= -(2\times 2\times 2\times 2\times z) + (2\times 2\times 5\times z\times z\times z)$$ $$= 2\times 2\times z[-(22\times 2\times z2)+(5\times z\times z)]$$ $$= 4z(-4+5z^2)$$ --- (v) $$20l^2m + 30alm$$ $$= (2\times 2\times 5\times l\times l\times m) + (2\times 3\times 5\times a\times l\times m)$$ $$= (2 \times 5 \times l\times m) [(2\times l) + (3 \times a)]$$ $$= 10 lm(2l + 3a)$$ --- (vi) $$5x^2y-15xy^2$$ $$=(5\times x\times x\times y)-(3\times 5\times x\times y\times y)$$ $$=5\times x\times y[x-(3\times y)]$$ $$= 5xy(x-3y)$$ --- (vii) $$10a^2 - 15b^2 + 20c^2 $$ $$ = 2\times 5\times a\times a-1\times 3\times 5\times b\times b + 2\times 2\times 5\times c\times c$$ $$ = 5 \times (2\times a\times a-1\times 3\times b\times b + 2\times 2\times c\times c)$$ $$ = 5(2a^2-3b^2+4c^2 )$$ --- (viii) $$-4a^2 + 4ab - 4ca $$ $$ = -1\times 2\times 2\times a\times a + 2\times 2\times a\times b -1\times 2\times 2\times c\times a $$ $$ = 2 \times 2 \times a(-a+b-c)$$ $$ = 4a(-a+b-c)$$ --- (ix) $$x^2yz + xy^2z + xyz^2$$ $$ = x\times x\times y\times z + x\times y\times y\times z + x\times y\times z\times z$$ $$= x\times y\times z(x+y+z)$$ $$= x\times y\times z(x+y+z)$$ --- (x) $$ax^2y + bxy^2 + cxyz$$ $$ = a\times x\times x\times y + b\times x\times y\times y + c\times x\times y\times z$$ $$ = x\times y\times (a\times x+b\times y+c\times z)$$ $$ = xy(ax+by+cz)$$

(i) $$x^2 + xy + 8x + 8y$$ $$ = x \times x + x \times y + 8 \times x + 8 \times y$$ $$ = x(x + y) + 8(x + y)$$ $$ = (x + 8)(x + y)$$ --- (ii) $$15xy-6x+5y-2 $$ $$ = 3 \times 5 \times x \times y - 2 \times 3 \times x + 5 \times y - 2 $$ $$ = 3x(5y-2)+1 (5y-2)$$ $$ = (5y-2) (3x+1)$$ --- (iii) $$ax + bx - ay- by$$ $$= a \times x + b \times x-a \times y-b \times y$$ $$= x(a+b) - y(a+b)$$ $$= (a + b) (x - y)$$ --- (iv) $$15pq+15+9q+ 25p$$ $$= 15pq + 9q + 25p + 15$$ $$= 3\times 5\times \times p\times q + 3\times 3 \times q + 5\times 5\times p + 3\times 5$$ $$= 3q (5p+3)+5 (5p+3)$$ $$= (5p+3) (3q+5)$$ --- (v) $$z- 7 + 7xy – xyz$$ $$= z - x\times y\times z - 7 + 7\times x\times y$$ $$= (1-xy) - 7(1-xy)$$ $$= (1 - xy )(z - 7)$$

(i) $$a^2 + 8a+ 16$$ $$= a^2+2\times 4\times a+4^2$$ $$= (a+4)2$$ --- (ii) $$p^2 - 10p+ 25$$ $$= p^2-2\times 5\times p+5^2$$ $$= (p-5)^2$$ --- (iii) $$25m^2 + 30m +9$$ $$= (5m)^2+2\times 5m\times 3+3^2$$ $$= (5m+3)^2$$ --- (iv) $$49y^2+84yz+36z^2$$ $$=(7y)^2+2\times 7y\times 6z+(6z)^2$$ $$= (7y+6z)^2$$ --- (v) $$4x^2 + 8x+4$$ $$= (2x)^2-2\times 4x+2^2$$ $$= (2x-2)^2$$ --- (vi) $$121b^2 - 88bc + 16c^2$$ $$= (11b)^2-2\times 11b\times 4c+(4c)^2$$ $$= (11b-4c)^2$$ --- (vii) $$(l + m)^2 - 4lm$$ (સૂચન : $$(l + m)^2$$ નું વિસ્તરણ કરો.) $$= l^2+m^2+2lm-4lm$$ $$= l^2+m^2-2lm$$ $$= (l-m)^2$$ --- (viii) $$a^4 + 2a^2b^2 + b^4$$ $$= (a^2)^2+2\times a^2\times b^2+(b^2)^2$$ $$= (a^2+b^2)^2$$

(i) $$4p^2 - 9q^2$$ $$= (2p)^2-(3q)^2$$ $$= (2p-3q)(2p+3q)$$ --- (ii) $$63a^2 - 112b^2$$ $$= 7(9a^2 –16b^2)$$ $$= 7((3a)^2–(4b)^2)$$ $$= 7(3a+4b)(3a-4b)$$ --- (iii) $$49x^2 - 36$$ $$= (7x)^2 -6^2$$ $$= (7x+6)(7x–6)$$ --- (iv) $$16x^5 - 144x^3$$ $$= 16x^3(x^2–9)$$ $$= 16x^3(x^2–9)$$ $$= 16x^3(x–3)(x+3)$$ --- (v) $$(l + m)^2 - (l − m)^2$$ $$= (l+m–l+m)(l+m+l–m)$$ $$= (2m)(2l)$$ $$= 4 ml$$ --- (vi) $$9x^2y^2 - 16$$ $$= (3xy)^2-4^2$$ $$= (3xy–4)(3xy+4)$$ --- (vii) $$(x^2 - 2xy + y^2) — z^2$$ $$= (x–y)^2–z^2$$ $$= {(x–y)–z}{(x–y)+z}$$ $$= (x–y–z)(x–y+z)$$ --- (viii) $$25a^2 - 4b^2 + 28bc - 49c^2 $$ $$= 25a^2–(4b2-28bc+49c2 )$$ $$= (5a)^2-{(2b)^2-2(2b)(7c)+(7c)^2}$$ $$= (5a)^2-(2b-7c)2$$ $$= (5a+2b-7c)(5a-2b+7c)$$

(i) $$ax^2 + bx$$ $$= x(ax+b)$$ --- (ii) $$7p^2 + 21q^2$$ $$ = 7(p^2+3q^2)$$ --- (iii) $$2x^3 + 2xy^2 + 2xz^2$$ $$ = 2x(x^2+y^2+z^2)$$ --- (iv) $$am^2 + bm^2 + bn^2 + an^2$$ $$= m^2(a+b)+n^2(a+b) $$ $$= (a+b)(m^2+n^2)$$ --- (v) $$(lm + l) + m + 1$$ $$= lm+m+l+1 $$ $$= m(l+1)+(l+1) $$ $$= (m+1)(l+1)$$ --- (vi) $$y(y + z) + 9(y + z)$$ $$= (y+9)(y+z)$$ --- (vii) $$5y^2 - 20y - 8z + 2yz $$ $$= 5y(y–4)+2z(y–4) $$ $$= (y–4)(5y+2z)$$ --- (viii) $$10ab + 4a + 5b + 2$$ $$= 5b(2a+1)+2(2a+1) $$ $$= (2a+1)(5b+2)$$ --- (ix) $$6xy - 4y + 6 - 9x$$ $$= 6xy–9x–4y+6 $$ $$= 3x(2y–3)–2(2y–3)$$ $$= (2y–3)(3x–2)$$

(i) $$a^4 - b^4$$ $$= (a^2)^2-(b^2)^2$$ $$= (a^2-b^2) (a^2+b^2)$$ $$= (a – b)(a + b)(a^2+b^2)$$ --- (ii) $$p^4 - 81$$ $$= (p^2)^2-(9)^2$$ $$= (p^2-9)(p^2+9)$$ $$= (p^2-3^2)(p^2+9)$$ $$=(p-3)(p+3)(p^2+9)$$ --- (iii) $$x^4 - (y + z)^4$$ $$= {x^2-(y+z)2}{ x^2+(y+z)^2}$$ $$= {(x –(y+z)(x+(y+z)}{x^2+(y+z)^2}$$ $$= (x–y–z)(x+y+z) {x^2+(y+z)^2}$$ --- (iv) $$x^4 − (x – z)^4$$ $$= {x^2-(x-z)^2}{x^2+(x-z)^2}$$ $$= { x-(x-z)}{x+(x-z)} {x^2+(x-z)^2}$$ $$= z(2x-z)( x^2+x^2-2xz+z^2)$$ $$= z(2x-z)( 2x^2-2xz+z^2)$$ --- (v) $$a^4 - 2a^2b^2 + b^4$$ $$= (a^2-b^2)^2$$ $$= ((a–b)(a+b))^2$$ $$= (a – b)^2 (a + b)^2$$

(i) $$p^2 + 6p + 8$$ $$= p^2+2p+4p+8 $$ $$= p(p+2)+4(p+2)$$ $$= (p+2)(p+4)$$ --- (ii) $$q^2 - 10q + 21$$ $$= q^2–3q-7q+21$$ $$= q(q–3)–7(q–3)$$ $$= (q–7)(q–3)$$ --- (iii) $$p^2 + 6p - 16$$ $$= p^2–2p+8p–16$$ $$= p(p–2)+8(p–2)$$ $$= (p+8)(p–2)$$

(i) $$28x^4 \div 56x$$ $$ = \frac{2 \times 2 \times 7 \times x \times x \times x \times x }{2 \times 2 \times 2 \times 7 \times x}$$ $$ = \frac{x^3}{2}$$ $$ = \frac{1}{2}x^3$$ --- (ii) $$-36y^3 \div 9y^2$$ $$ = \frac{ - 2 \times 2 \times 3 \times 3 \times y \times y \times y }{3 \times 3 \times y \times y }$$ $$ = -4y$$ --- (iii) $$66pq^2r^3 \div 11qr^2$$ $$ = \frac{2 \times 3 \times 11\times p\times q \times q \times r\times r\times r }{11 \times q \times r \times r }$$ $$ = 6pqr$$ --- (iv) $$34x^3y^3z^3 \div 51xy^2z^3$$ $$ = \frac{2 \times 17 \times x \times x \times x \times y \times y \times y \times z \times z\times z}{3 \times 17 \times x \times y \times z\times z \times z}$$ $$ = \frac{2}{3}x^2y$$ --- (v) $$12a^8b^8 \div (-6a^6b^4)$$ $$ = \frac{ 2 \times 2 \times 3 \times a^6 \times a^2\times b^4 \times b^4}{-2 \times 3 \times a^6 \times b^4}$$ $$ = -2a^2b^4$$

(i) $$(5x^2-6x) \div 3x$$ $$ = \frac{x(5x - 6)}{3x}$$ $$ = \frac{1}{3}(5x - 6)$$ --- (ii) $$(3y^8–4y^6+5y^4) \div y^4$$ $$ = \frac{y^4(3y^4 - 4y^2 + 5)}{y^4}$$ $$ = 3y^4 - 4y^2 + 5$$ --- (iii) $$8(x^3y^2z^2+x^2y^3z^2+x^2y^2z^3) \div 4x^2 y^2 z^2$$ $$ = \frac{8x^2 y^2 z^2(x + y + z)}{4x^2 y^2 z^2}$$ $$ = 2(x + y + z)$$ --- (iv) $$(x^3+2x^2+3x) \div 2x$$ $$ = \frac{x(x^2+2x+3)}{2x}$$ $$ = \frac{1}{2}(x^2+2x+3)$$ --- (v) $$(p^3q^6–p^6q^3) \div p^3q^3$$ $$ = \frac{p^3q^3(p^3 - q^3)}{p^3q^3}$$ $$ = p^3 - q^3$$

(i) $$(10x–25) \div 5$$ $$ = \frac{5(2x - 5)}{5}$$ $$ = 2x - 5$$ --- (ii) $$(10x–25) \div (2x–5)$$ $$ = \frac{5(2x–5)}{(2x–5)}$$ $$ = 5$$ --- (iii) $$10y(6y+21) \div 5(2y+7)$$ $$ = \frac{10y\times 3 (2y+7)}{5(2y+7)}$$ $$ = 6y$$ --- (iv) $$9x^2y^2(3z–24) \div 27xy(z–8)$$ $$ = \frac{9x^2y^2\times 3(z–8)}{27xy(z–8)}$$ $$ = xy$$ --- (v) $$96abc(3a - 12)(5b - 30) \div 144(a - 4) (b - 6) $$ $$ = \frac{96abc\times 3(a - 4) \times 5(b - 6)}{144(a - 4) (b - 6)}$$ $$ = 10abc$$

(i) $$5(2x+1)(3x+5) \div (2x+1)$$ $$ = \frac{5(2x+1)(3x+5)}{(2x+1)}$$ $$ = 5(3x+5)$$ --- (ii) $$26xy(x+5)(y–4)\div 13x(y–4)$$ $$ = \frac{26xy(x+5)(y–4)}{13x(y–4)}$$ $$ = 2y(x+5)$$ --- (iii) $$52pqr(p+q)(q+r)(r+p) \div 104pq(q+r)(r+p)$$ $$ = \frac{52pqr(p+q)(q+r)(r+p)}{104pq(q+r)(r+p)}$$ $$ = \frac{1}{2}r(p+q)$$ --- (iv) $$20(y+4) (y^2+5y+3) \div 5(y+4)$$ $$ = \frac{20(y+4) (y^2+5y+3) }{5(y+4)}$$ $$ = 4(y^2+5y+3)$$ --- (v) $$x(x+1)(x+2)(x+3) \div x(x+1)$$ $$ = \frac{x(x+1)(x+2)(x+3)}{x(x+1)}$$ $$ = (x+2)(x+3)$$

(i) $$(y^2+7y+10)\div (y+5)$$ $$ = \frac{(y^2+7y+10)}{(y+5)}$$ $$ = \frac{(y+2)(y+5)}{(y+5)}$$ $$ = y + 2$$ --- (ii) $$(m^2–14m–32)\div (m+2)$$ $$ = \frac{(m^2–14m–32)}{(m+2)}$$ $$ = \frac{(m–16)(m+2)}{(m+2)}$$ $$ = m - 16$$ --- (iii) $$(5p^2–25p+20)\div (p–1)$$ $$ = \frac{(5p^2–25p+20)}{(p–1)}$$ $$ = \frac{5(p–1)(p–4)}{(p–1)}$$ $$ = 5(p-4)$$ --- (iv) $$4yz(z^2+6z–16)\div 2y(z+8)$$ $$ = \frac{4yz(z^2+6z–16)}{2y(z+8)}$$ $$ = \frac{4yz(z–2)(z+8)}{2y(z+8)}$$ $$ = 2z(z-2)$$ --- (v) $$5pq(p^2–q^2)\div 2p(p+q)$$ $$ = \frac{5pq(p^2–q^2)}{2p(p+q)}$$ $$ = \frac{5pq(p–q)(p+q)}{2p(p+q)}$$ $$ = \frac{5}{2}q(p–q)$$ --- (vi) $$12xy(9x^2–16y^2)\div 4xy(3x+4y)$$ $$ = \frac{12xy(9x^2–16y^2)}{4xy(3x+4y)}$$ $$ = \frac{12xy(3x+4y)(3x-4y)}{4xy(3x+4y)}$$ $$ = 3(3x-4y)$$ --- (vii) $$39y^3(50y^2–98) \div 26y^2(5y+7)$$ $$ = \frac{39y^3(50y^2–98)}{26y^2(5y+7)}$$ $$ = \frac{39y^3 \times 2(5y–7)(5y+7)}{26y^2(5y+7)}$$ $$ = 3y(5y - 7)$$