प्रश्नावली-8A
1.दिए गए पदों में सार्वगुणनखंड ज्ञात कीजिए
(a)27x और 15
27x=3×3×x
15=3×5
सार्वगुणनखंड=3
(b)35a2b और 28ab2
35a2b=7×5×a×a×b
28ab2=7×2×2×a×b×b
सार्वगुणनखंड=7×a×b=7ab
(c)42ab और 56a2
42ab=2×3×7×a×b
56a2=2×2×2×7×a×a
सार्वगुणनखंड=2×7a=14a
(d)12ab, 18a2b और 24abc
12ab=2×2×3×a×b
18a2b=2×3×3×a×a×b
24abc=2×2×2×3abc
सार्वगुणनखंड=2×3ab=6ab
(e)5p, 10p2 और 15
5p=5×p
10p2=2×5×p×p
15=3×5
सार्वगुणनखंड=5
(f)6p2q3, 30p3q2 और -12p2q2
6p2q3=2×3×p×p×q×q×q
30p3q2=2×3×5×p×p×p×q×q
-12p2q2=-2×2×3×p×p×q×q
सार्वगुणनखंड=2×3×p×p×q×q=6p2q2
(g)8a3b3, 12a2b, 20ab2
8a3b3=2×2×2×a×a×a×b×b×b
20ab2=2×2×5×a×b×b
सार्वगुणनखंड=2×2×a×b=4ab
(h)-32m2, 64m, 16m2n
-32m2=-2×2×2×2×2×m×m
64m=2×2×2×2×2×2×m
सार्वगुणनखंड=2×2×2×2m=16m
2. निम्नलिखित व्यंजकों के गुणनखण्ड कीजिए-
(a)6x+9
=3(2x+3)
(b)12a-16
=4(3a-4)
(c)8×2+12x
4x(2x+3)
(d)-32p+40p3
8p(-4+5p2)
(e) 10a2b-15abc
=5ab(2a-3c)
(f)6x2y+15xy2
=3xy(2x+5y)
(g) 15a2+25b2-30ab
=5(3a2+5b2-6ab)
(h)-18a2+12ab-24ac
=6a(-3a+2b-4c)
(i)a3b2c2-ab2c3+a2b3c
=ab2c(a2c-c2+ab)
(j)3a2b3-15a3b2-6a2b2c2
=3a2b2(b-5a-2c2)
3. गुणनखण्ड कीजिए-
(a)m2+mn-7m-7n
=m(m+n)-7(m+n)
=(m+n)(m-7)
(b)9ab-6a+12b-8
=3a(3b-2)+4(3b-2)
=(3a+4)(3b-2)
(c)3pq+12p-2q-8
=3p(q+4)-2(q+4)
=(q+4)3p-2)
(d)2a2+3a-2ab-3b
=a(2a+3)-b(2a+3)
=(a-b)(2a+3)
(e)m2+2mn-3m-6n
=m(m+2n)-3(m+2n)
=(m+2n)(m-3)
(f)6ax-9bx+8ay-12by
=2a(3x+4y)-3b(3x+4y)
=(2a-3b)(3x+4y)
(g)-ab+6c-ac+6b
=-a(b+c)+6(b+c)
=(b+c)(-a+6)
(h)8a-7b-56+ab
=8(a-7)+b(-7+a)
=(b+8)(a-7)
(i)xy+56-7y-8x
=y(x-7)-8(x-7)
=(x-7)(y-8)
(j)ac+2b-a-2bc
=ac-2bc+2b-a
=c(a-2b)-1(a-2b)
=(c-1)(a-2b)
4. गुणनखंड कीजिए—
(a)a2+18a+81
=a2+9a+9a+81
=a(a+9)+9(a+9)
=(a+9)(a+9)
(b)m2-10m+25
=m2-5m-5m+25
=m(m-5)-5(m-5)
=(m-5)(m-5)
(c)2×2+16x+32
=2×2+8x+8x+32
=2x(x+4)+8(x+4)
=(2x+8)(x+4)
(d)3y2-30y+75
=3y2-15y-15y+75
=3y(y-5)-15(y-5)
=(y-5)(3y-15)
=3(y-5)(y-5)
(e)(2x+y)2-8xy
=(2x)2+2×2x×y+(y)2-8xy
=4×2+4xy+y2-8xy
=4×2-4xy+y2
=(2x-y)2
(f)(a-3b)2+12ab
=(a)2-2×a×3b+(3b)2+12ab
=a2-6ab+9b2+12ab
=a2+6ab+9b2
=(a+3b)2
(g)9p4+12p2q2+4q4
=(3p2)2+2×3p2×2q2+(2q2)2
=(3p2+2q2)2
(h)4m2-28mn+49n2
=4m2-14mn-14mn+49n2
=2m(2m-7n)-7n(2m-7n)
=(2m-7n)(2m-7n)
5. गुणनखण्ड निकालिए-
(a)9a2 – 4b2
=(3a)2-(2b)2
=(3a+2b)(3a-2b)
(b)49a2-36b2
=(7a)2-(6b)2
=(7a+6b)(7a-6b)
(c)98p2-72
=2(49p2 – 36)
=2{(7p)2-(6)2}
=2(7p+6)(7p-6)
(d)121a5 – 169a3
=a3(121a2-169)
=a3{(11a)2-(13)2}
=a3(11a+13)(11a-13)
(e)18a2b2-32
=2(9a2b2-16)
=2{(3ab)2-(4)2}
=2(3ab+4)(3ab-4)
(f)(2a+b)2-(2a-b)2
=(2a+b-2a+b)(2a+b+2a-b)
=2b×4a=8ab
(g)9a2-12ab+4b2-16c2
=(9a2-12ab+4b2)-16c2
={(3a)2-2×3a×2b+(2b)2}-(4c)2
={(3a-2b)2}-(4c)2
=(3a-2b+4c)(3a-2b-4c)
(h)49a2-25b2+14a-10b
={(7a)2-(5b)2}+2(7a-5b)
=(7a-5b)(7a+5b)+2(7a-5b)
=(7a-5b)(7a+5b+2)
(i)a2-b2+2bc-c2
=a2-(b2-2bc+c2)
=(a)2-{(b)2-2×b×c+(c)2}
=(a)2-{(b-c)2}
=(a-b+c)(a+b-c)
(j)4×2-4y2-4yz-z2
=4×2-(4y2+4yz+z2)
=(2x)2-{(2y)2+2×2y×z+(z)2}
=(2x)2-(2y+z)2
=(2x+2y+z)(2x-2y-z)
(k)a4 – b4
=(a2)2 – (b2)2
=(a2+b2)(a2-b2)
=(a2+b2)(a+b)(a-b)
(l)m4-256
=(m2)2-(16)2
=(m2+16)(m2-16)
=(m2+16){(m2-(4)2}
=(m2+16)(m2+4)(m2-4)
(m)(a+b)4 – c4
={(a+b)2 – (c2)2}
={(a+b)2+c2{(a+b)2-(c)2}
={(a+b)2+c2}(a+b+c)(a+b-c)
(n)y4 – (2+x)4
=(y2)2 – {(2+x)2}2
={y2-(2+x)2}(y2+(2+x)2}
=(y+2+x)(y-2-x)(y2+(2+x)2}
(o)(a+b)4 – (a-b)4
={(a+b)2}2 – {(a-b)2}2
={(a+b)2+(a-b)2}{(a+b)2-(a-b)2}
=(a2+2ab+b2+a2-2ab+b2)(a2+b2+2ab-a2+2ab-b2)
=(2a2+2b2)(4ab)
=2(a2+b2)×4ab=8ab(a2+b2)
6. गुणनखंडन कीजिए—
(a)x2+11x+28
=x2+7x+4x+28
=x(x+7)+4(x+7)
=(x+4)(x+7)
(b)x2+13x+42
=x2+6x+7x+42
=x(x+6)+7(x+6)
=(x+6)(x+7)
(c)a2+17a+72
=a2+8a+9a+72
=a(a+8)+9(a+8)
=(a+8)(a+9)
(d)p2+16p+63
=p2+7p+9p+63
=p(p+7)+9(p+7)
=(p+7)(p+9)
(e) x2-11x+30
=x2-6x-5x+30
=x(x-6)-5(x-6)
=(x-6)(x-5)
(f)y2-12y+20
=y2-10y-2y+20
=y(y-10)-2(y-10)
=(y-10)(y-2)
(g)x2-22x+85
=x2-17x-5x+85
=x(x-17)-5(x-17)
=(x-17)(x-5)
(h)x2-19x+60
=x2-15x-4x+60
=x(x-15)-4(x-15)
=(x-15)(x-4)
(i)x2-3x-28
=x2+4x-7x-28
=x(x+4)-7(x+4)
=(x+4)(x-7)
(j)m2-6m-40
=m2-10m+4m-40
=m(m-10)+4(m-10)
=(m-10)(m+4)
(k)x2-11x-60
=x2-15x+4x-60
=x(x-15)+4(x-15)
=(x+4)(x-15)
(l)a2-a-12
=a2+3a-4a-12
=a(a+3)-4(a+3)
=(a+3)(a-4)
(m)y2-5y-24
=y2+3y-8y-24
=y(y+3)-8(y+3)
=(y+3)(y-8)
(n)z2-2z-24
=z2-6z+4z-24
=z(z-6)+4(z-6)
=(z-6)(z+4)
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