प्रश्नावली-14.2
1. निम्नलिखित व्यंजकों का गुणनफल ज्ञात कीजिए:
(a)1+2x+x2 (b)a2b2-6abc+9c2
(c)1-(a-b)2 (d)16(a-b)2-9(a+b)2
(e)(x+y)2-10(x+y)+25 (f)(a+b)2-4ab
(g)4×2-y2+4y-4 (h)9×2 – n2
4
(i)a2+a+4+3a (j)x2+6x+8
(k)y2-13y+30 (l)x2+9x-22
उत्तर:—
(a)1+2x+x2=(1)2+2×1×x+(x)2
=(1+x)2=(1+x)(1+x)
(b)a2b2-6abc+9c2
=(ab)2-2×ab×3c+(3c)2
=(ab-3c)2=(ab-3c)(ab-3c)
(c)1-(a-b)2=(1)2-(a-b)2
=(1+a-b)(1-(a-b)=(1+a-b)(1-a+b)
(d)16(a-b)2-9(a+b)2
={4(a-b)}2-{3(a+b)}2
={4(a-b)+3(a+b)}{(4(a-b)-3(a+b)}
=(4a-4b+3a+3b)(4a-4b-3a-3b)
=(7a-b)(a-7b)
(e)(x+y)2-10(x+y)+25
=(x+y)2-2×5×(x+y)+(5)2
=(x+y-5)2=(x+y-5)(x+y-5)
(f)(a+b)2-4ab=a2+b2+2ab-4ab
=a2+b2-2ab=(a-b)2=(a-b)(a-b)
(g)4×2-y2+4y-4=4×2-(y2-4y+4)
=(2x)2-(y-2)2=(2x+y-2)(2x-y+2)
(h)9×2 – n2
4
(3x)2-( n )2
2
(3x+ n )(3x- n )
2 2
(i)a2+a+4+3a=a2+4a+4
=(a)2+2×a×2+(2)2=(a+2)2=(a+2)(a+2)
(j)x2+6x+8=x2+6x+9-1
=(x)2+2×x×3+(3)2-(1)2
=(x+3)2-(1)2=(x+3+1)(x+3-1)
=(x+4)(x+2)
(k)y2-13y+30=y2-3y-10y+30
=y(y-3)-10(y-3)=(y-10)(y-3)
(l)x2+9x-22=x2+11x-2x-22
=x(x+11)-2(x+11)=(x-2)(x+11)
2.निम्नलिखित व्यंजकों का गुणनखण्ड कीजिए
(a)x2-6x-135 (b)8(x+y)3-50(x+y)
(c)4×2+9y2+12xy-1 (d)75-x2+10x
(e)12a2-27 (f)ax2-bx2+by2-ay2
उत्तर:—
(a)x2-6x-135=x2-15x+9x-135
=x(x-15)+9(x-15)=(x+9)(x-15)
(b)8(x+y)3-50(x+y)
=2(x+y){4(x+y)2-25}
=2(x+y){(2(x+y)2-(5)2}
=2(x+y){2(x+y)+5)(2(x+y)-5)}
=2(x+y)(2x+2y+5)(2x+2y-5)
(c)4×2+9y2+12xy-1
=(2x)2+2×2x×3y+(3y)2-(1)2
=(2x+3y)2-(1)2
=(2x+3y+1)(2x+3y-1)
(d)75-x2+10x
-(x2-10x-75)
-(x2-15x+5x-75)
-{x(x-15)+5(x-15)}
-(x-15)(x+5)
(15-x)(5+x)
(e)12a2-27
3(4a2-9)
3{(2a)2-(3)2}
3(2a+3)(2a-3)
(f)ax2-bx2+by2-ay2
=ax2-ay2-bx2+by2
=a(x2-y2)-b(x2-y2)
(x2-y2)(a-b)
(x+y)(x-y)(a-b)
(a-b)(x+y)(x-y)
3.निम्नलिखित व्यंजकों का गुणनखंडन
कीजिए
(a) 16×4-81y4 (b)x4 – 1
(c)x4 – (x-y)4 (d)9×2-4y2-3x+2y
(e)(x+y)3+4(x+y)2+4x+4y
उत्तर:—
(a) 16×4-81y4= (4×2)2-(9y2)2
(4×2+9y2)(4×2-9y2)
(4×2+9y2){(2x)2-(3y)2}
(4×2+9y2)(2x+3y)(2x-3y)
(b)x4 – 1
(x2)2 – (1)2
(x2-1)(x2+1)
(x+1)(x-1)(x2+1)
(c)x4 – (x-y)4
(x2)2-(x-y)2
(x2+x-y)(x2-x+y)
(d)9×2-4y2-3x+2y
(3x)2-(2y)2-(3x-2y)
(3x+2y)(3x-2y)-(3x-2y)
(3x-2y)(3x+2y-1)
(e)(x+y)3+4(x+y)2+4x+4y
(x+y)3+4(x+y)2+4(x+y)
(x+y)[(x+y)2+4(x+y)+4]
(x+y)(x+y+2)2
(x+y)(x+y+2)(x+y+2)
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