অষ্টম শ্রেণি – অধ্যায় ১৪ : বীজগাণিতিক সংখ্যামালার গ.সা.গু. ও ল.সা.গু সম্পূর্ণ সমাধান

কষে দেখি – 14

1. নীচের বীজগাণিতিক সংখ্যামালাগুলির গ.সা.গু. নির্ণয় করি –

\left( i \right)4{{a}^{2}}{{b}^{2}},20a{{b}^{2}}          \left( ii \right)5{{p}^{2}}{{q}^{2}},10{{p}^{2}}{{q}^{2}},25{{p}^{4}}{{q}^{3}}
\left( iii \right)7{{y}^{3}}{{z}^{6}},21{{y}^{2}},14{{z}^{2}}          \left( iv \right)3{{a}^{2}}{{b}^{2}}c,12{{a}^{2}}{{b}^{4}}{{c}^{2}},9{{a}^{5}}{{b}^{4}}

উত্তর-

(i)

প্রথম সংখ্যামালা, 4{{a}^{2}}{{b}^{2}}=2\times 2\times a\times a\times b\times b
দ্বিতীয় সংখ্যামালা, 20a{{b}^{2}}=2\times 2\times 5\times a\times b\times b
∴ নির্ণেয় গ.সা.গু. = 2\times 2\times a\times b\times b=4a{{b}^{2}}

(ii)

প্রথম সংখ্যামালা, 5{{p}^{2}}{{q}^{2}}=5\times p\times p\times q\times q
দ্বিতীয় সংখ্যামালা,10{{p}^{2}}{{q}^{2}}=2\times 5\times p\times p\times q\times q
তৃতীয় সংখ্যামালা,25{{p}^{4}}{{q}^{3}}=5\times 5\times p\times p\times p\times p\times q\times q\times q
∴ নির্ণেয় গ.সা.গু. = 5\times p\times p\times q\times q=5{{p}^{2}}{{q}^{2}}

(iii)

প্রথম সংখ্যামালা, 7{{y}^{3}}{{z}^{6}}=7\times y\times y\times y\times z\times z\times z\times z\times z\times z
দ্বিতীয় সংখ্যামালা,21{{y}^{2}}=3\times 7\times y\times y
তৃতীয় সংখ্যামালা,14{{z}^{2}}=2\times 7\times z\times z
∴ নির্ণেয় গ.সা.গু. = 7

(iv)

প্রথম সংখ্যামালা, 3{{a}^{2}}{{b}^{2}}c=3\times a\times a\times b\times b\times c
দ্বিতীয় সংখ্যামালা,12{{a}^{2}}{{b}^{4}}{{c}^{2}}=2\times 2\times 3\times a\times a\times b\times b\times b\times b\times c\times c
তৃতীয় সংখ্যামালা, 9{{a}^{5}}{{b}^{4}}=3\times 3\times a\times a\times a\times a\times a\times b\times b\times b\times b
∴ নির্ণেয় গ.সা.গু. = 3\times a\times a\times b\times b=3{{a}^{2}}{{b}^{2}}

2. নীচের বীজগাণিতিক সংখ্যামালাগুলির ল.সা.গু. নির্ণয় করি –

\left( i \right)2{{x}^{3}}{{y}^{3}},10{{x}^{3}}y          \left( ii \right)7{{p}^{2}}{{q}^{3}},35{{p}^{3}}q,42p{{q}^{4}}
\left( iii \right)5{{a}^{5}}b,15a{{b}^{2}}c,25{{a}^{2}}{{b}^{2}}{{c}^{2}}          \left( iv \right)11{{a}^{2}}b{{c}^{2}},33{{a}^{2}}{{b}^{2}}c,55{{a}^{2}}b{{c}^{2}}

উত্তর-

(i)

প্রথম সংখ্যামালা, 2{{x}^{3}}{{y}^{3}}=2\times x\times x\times x\times y\times y\times y
দ্বিতীয় সংখ্যামালা, 10{{x}^{3}}y=2\times 5\times x\times x\times x\times y
∴ নির্ণেয় ল.সা.গু.= 2\times 5\times x\times x\times x\times y\times y\times y=10{{x}^{3}}{{y}^{3}}

(ii)

প্রথম সংখ্যামালা, 7{{p}^{2}}{{q}^{3}}=7\times p\times p\times q\times q\times q
দ্বিতীয় সংখ্যামালা,35{{p}^{3}}q=5\times 7\times p\times p\times p\times q
তৃতীয় সংখ্যামালা,42p{{q}^{4}}=2\times 3\times 7\times p\times q\times q\times q\times q
∴ নির্ণেয় ল.সা.গু. = 2\times 3\times 5\times 7\times p\times p\times p\times q\times q\times q\times q=210{{p}^{3}}{{q}^{4}}

(iii)

প্রথম সংখ্যামালা, 5{{a}^{5}}b=5\times a\times a\times a\times a\times a\times b
দ্বিতীয় সংখ্যামালা,15a{{b}^{2}}c=3\times 5\times a\times b\times b\times c
তৃতীয় সংখ্যামালা,25{{a}^{2}}{{b}^{2}}{{c}^{2}}=5\times 5\times a\times a\times b\times b\times c\times c
∴ নির্ণেয় ল.সা.গু. = 3\times 5\times 5\times a\times a\times a\times a\times a\times b\times b\times c\times c=75{{a}^{5}}{{b}^{2}}{{c}^{2}}

(iv)

প্রথম সংখ্যামালা, 11{{a}^{2}}b{{c}^{2}}=11\times a\times a\times b\times c\times c
দ্বিতীয় সংখ্যামালা,33{{a}^{2}}{{b}^{2}}c=3\times 11\times a\times a\times b\times b\times c
তৃতীয় সংখ্যামালা, 55{{a}^{2}}b{{c}^{2}}=5\times 11\times a\times a\times b\times c\times c
∴ নির্ণেয় ল.সা.গু. = 3\times 5\times 11\times a\times a\times b\times b\times c\times c=165{{a}^{2}}{{b}^{2}}{{c}^{2}}

3. নীচের বীজগাণিতিক সংখ্যামালাগুলির গ.সা.গু. নির্ণয় করি –

(i)5x(x+y),{{x}^{3}}-x{{y}^{2}}          (ii){{x}^{3}}-3{{x}^{2}}y,{{x}^{2}}-9{{y}^{2}}
(iii)2ax{{(a-x)}^{2}},4{{a}^{2}}x{{(a-x)}^{3}}          \left( iv \right){{x}^{2}}-1,{{x}^{2}}-2x+1,{{x}^{3}}+{{x}^{2}}-2x
\left( v \right){{a}^{2}}-1,{{a}^{3}}-1,{{a}^{2}}+a-2          \left( vi \right){{x}^{2}}+3x+2,{{x}^{2}}+4x+3,{{x}^{2}}+5x+6
\left( vii \right){{x}^{2}}+xy,xz+yz,{{x}^{2}}+2xy+{{y}^{2}}          \left( viii \right)8\left( {{x}^{2}}-4 \right),12\left( {{x}^{3}}+8 \right),36\left( {{x}^{2}}-3x-10 \right)
\left( ix \right){{a}^{2}}-{{b}^{2}}-{{c}^{2}}+2bc,{{b}^{2}}-{{c}^{2}}-{{a}^{2}}+2ac,{{c}^{2}}-{{a}^{2}}-{{b}^{2}}+2ab          \left( x \right){{x}^{3}}-16x,2{{x}^{3}}+9{{x}^{2}}+4x,2{{x}^{3}}+{{x}^{2}}-28x
\left( xi \right)4{{x}^{2}}-1,8{{x}^{3}}-1,4{{x}^{2}}-4x+1          \left( xii \right){{x}^{3}}-3{{x}^{2}}-10x,{{x}^{3}}+6{{x}^{2}}+8x,{{x}^{4}}-5{{x}^{3}}-14{{x}^{2}}
\left( xiii \right)6{{x}^{2}}-13xa+6{{a}^{2}},6{{x}^{2}}+11xa-10{{a}^{2}},6{{x}^{2}}+2xa-4{{a}^{2}}

উত্তর-

(i)

প্রথম সংখ্যামালা, 5x(x+y)=5\times x(x+y)
দ্বিতীয় সংখ্যামালা, {{x}^{3}}-x{{y}^{2}}=x\left( {{x}^{2}}-{{y}^{2}} \right)=x\left( x+y \right)\left( x-y \right)
∴ নির্ণেয় গ.সা.গু. = x\left( x+y \right)

(ii)

প্রথম সংখ্যামালা, {{x}^{3}}-3{{x}^{2}}y={{x}^{2}}\left( x-3y \right)=x\times x\left( x-3y \right)
দ্বিতীয় সংখ্যামালা,{{x}^{2}}-9{{y}^{2}}={{\left( x \right)}^{2}}-{{\left( 3y \right)}^{2}}=\left( x+3y \right)\left( x-3y \right)
∴ নির্ণেয় গ.সা.গু. = \left( x-3y \right)

(iii)

প্রথম সংখ্যামালা, 2ax{{(a-x)}^{2}}=2\times a\times x\times \left( a-x \right)\left( a-x \right)
দ্বিতীয় সংখ্যামালা,4{{a}^{2}}x{{(a-x)}^{3}}=2\times 2\times a\times a\times x\times \left( a-x \right)\left( a-x \right)\left( a-x \right)
∴ নির্ণেয় গ.সা.গু. = 2\times a\times x\times \left( a-x \right)\left( a-x \right)=2ax{{(a-x)}^{2}}

(iv)

প্রথম সংখ্যামালা, {{x}^{2}}-1=\left( x+1 \right)\left( x-1 \right)
দ্বিতীয় সংখ্যামালা,{{x}^{2}}-2x+1={{x}^{2}}-2.x.1+{{1}^{2}}={{\left( x-1 \right)}^{2}}=\left( x-1 \right)\left( x-1 \right)
তৃতীয় সংখ্যামালা, {{x}^{3}}+{{x}^{2}}-2x=x\left\{ {{x}^{2}}+x-2 \right\}=x\left\{ {{x}^{2}}+\left( 2-1 \right)x-2 \right\}
=x\left\{ {{x}^{2}}+2x-x-2 \right\}=x\left\{ x\left( x+2 \right)-1\left( x+2 \right) \right\}=x\left( x+2 \right)\left( x-1 \right)
∴ নির্ণেয় গ.সা.গু. = \left( x-1 \right)

(v)

প্রথম সংখ্যামালা, {{a}^{2}}-1=\left( a+1 \right)\left( a-1 \right)
দ্বিতীয় সংখ্যামালা, {{a}^{3}}-1=\left( a-1 \right)\left( {{a}^{2}}+a+1 \right)
তৃতীয় সংখ্যামালা,{{a}^{2}}+a-2
={{a}^{2}}+\left( 2-1 \right)a-2
={{a}^{2}}+2a-a-2
=a\left( a+2 \right)-1\left( a+2 \right)
=\left( a+2 \right)\left( a-1 \right)
∴ নির্ণেয় গ.সা.গু. = \left( a-1 \right)

(vi)

প্রথম সংখ্যামালা, {{x}^{2}}+3x+2

\[={{x}^{2}}+\left( 2+1 \right)x+2\]

\[={{x}^{2}}+2x+x+2\]

\[=x\left( x+2 \right)+1\left( x+2 \right)\]

\[=\left( x+2 \right)\left( x+1 \right)\]

দ্বিতীয় সংখ্যামালা,{{x}^{2}}+4x+3

\[={{x}^{2}}+\left( 3+1 \right)x+3\]

\[={{x}^{2}}+3x+x+3\]

\[=x\left( x+3 \right)+1\left( x+3 \right)\]

\[=\left( x+3 \right)\left( x+1 \right)\]

তৃতীয় সংখ্যামালা,{{x}^{2}}+5x+6

\[={{x}^{2}}+\left( 3+2 \right)x+6\]

\[={{x}^{2}}+3x+2x+6\]

\[=x\left( x+3 \right)+2\left( x+3 \right)\]

\[=\left( x+3 \right)\left( x+2 \right)\]

∴ নির্ণেয় গ.সা.গু. = 1

(vii)

প্রথম সংখ্যামালা, {{x}^{2}}+xy=x\left( x+y \right)
দ্বিতীয় সংখ্যামালা,xz+yz=z\left( x+y \right)
তৃতীয় সংখ্যামালা,{{x}^{2}}+2xy+{{y}^{2}}={{\left( x+y \right)}^{2}}=\left( x+y \right)\left( x+y \right)
∴ নির্ণেয় গ.সা.গু. = \left( x+y \right)

(viii)

প্রথম সংখ্যামালা, 8\left( {{x}^{2}}-4 \right)

\[=2\times 2\times 2\left\{ {{\left( x \right)}^{2}}-{{\left( 2 \right)}^{2}} \right\}\]

\[=2\times 2\times 2\left( x+2 \right)\left( x-2 \right)\]

দ্বিতীয় সংখ্যামালা,12\left( {{x}^{3}}+8 \right)

\[=2\times 2\times 3\left\{ {{\left( x \right)}^{3}}+{{\left( 2 \right)}^{3}} \right\}\]

\[=2\times 2\times 3\left( x+2 \right)\left( {{x}^{2}}-2x+4 \right)\]

তৃতীয় সংখ্যামালা, 36\left( {{x}^{2}}-3x-10 \right)

\[=2\times 2\times 3\times 3\left\{ {{x}^{2}}-\left( 5-2 \right)x-10 \right\}\]

\[=2\times 2\times 3\times 3\left\{ {{x}^{2}}-5x+2x-10 \right\}\]

\[=2\times 2\times 3\times 3\left\{ x\left( x-5 \right)+2\left( x-5 \right) \right\}\]

\[=2\times 2\times 3\times 3\left( x-5 \right)\left( x+2 \right)\]

∴ নির্ণেয় গ.সা.গু. = 2\times 2\left( x+2 \right)=4\left( x+2 \right)

(ix)

প্রথম সংখ্যামালা, {{a}^{2}}-{{b}^{2}}-{{c}^{2}}+2bc

\[={{a}^{2}}-\left\{ {{b}^{2}}+{{c}^{2}}-2bc \right\}\]

\[={{a}^{2}}-{{\left( b-c \right)}^{2}}\]

\[=\left( a+b-c \right)\left( a-b+c \right)\]

দ্বিতীয় সংখ্যামালা, {{b}^{2}}-{{c}^{2}}-{{a}^{2}}+2ac

\[={{b}^{2}}-\left\{ {{c}^{2}}+{{a}^{2}}-2ac \right\}\]

\[={{b}^{2}}-{{\left( c-a \right)}^{2}}\]

\[=\left( b+c-a \right)\left( b-c+a \right)\]

তৃতীয় সংখ্যামালা,{{c}^{2}}-{{a}^{2}}-{{b}^{2}}+2ab

\[={{c}^{2}}-\left\{ {{a}^{2}}+{{b}^{2}}-2ab \right\}\]

\[={{c}^{2}}-{{\left( a-b \right)}^{2}}\]

\[=\left( c+a-b \right)\left( c-a+b \right)\]

∴ নির্ণেয় গ.সা.গু. = 1

(x)

প্রথম সংখ্যামালা, {{x}^{3}}-16x

\[=x\left( {{x}^{2}}-16 \right)\]

\[=x\left\{ {{\left( x \right)}^{2}}-{{\left( 4 \right)}^{2}} \right\}\]

\[=x\left( x+4 \right)\left( x-4 \right)\]

দ্বিতীয় সংখ্যামালা,2{{x}^{3}}+9{{x}^{2}}+4x

\[=x\left\{ 2{{x}^{2}}+9x+4 \right\}\]

\[=x\left\{ 2{{x}^{2}}+\left( 8+1 \right)x+4 \right\}\]

\[=x\left\{ 2{{x}^{2}}+8x+x+4 \right\}\]

\[=x\left\{ 2x\left( x+4 \right)+1\left( x+4 \right) \right\}\]

\[=x\left( x+4 \right)\left( 2x+1 \right)\]

তৃতীয় সংখ্যামালা,2{{x}^{3}}+{{x}^{2}}-28x

\[=x\left\{ 2{{x}^{2}}+x-28 \right\}\]

\[=x\left\{ 2{{x}^{2}}+\left( 8-7 \right)x-28 \right\}\]

\[=x\left\{ 2{{x}^{2}}+8x-7x-28 \right\}\]

\[=x\left\{ 2x\left( x+4 \right)-7\left( x+4 \right) \right\}\]

\[=x\left( x+4 \right)\left( 2x-7 \right)\]

∴ নির্ণেয় গ.সা.গু. = x\left( x+4 \right)

(xi)

প্রথম সংখ্যামালা, 4{{x}^{2}}-1

\[={{\left( 2x \right)}^{2}}-{{\left( 1 \right)}^{2}}\]

\[=\left( 2x+1 \right)\left( 2x-1 \right)\]

দ্বিতীয় সংখ্যামালা,8{{x}^{3}}-1

\[={{\left( 2x \right)}^{3}}-{{\left( 1 \right)}^{3}}\]

\[=\left( 2x-1 \right)\left( 4{{x}^{2}}+2x+1 \right)\]

তৃতীয় সংখ্যামালা, 4{{x}^{2}}-4x+1

\[={{\left( 2x \right)}^{2}}-2.2x.1+{{\left( 1 \right)}^{2}}\]

\[={{\left( 2x-1 \right)}^{2}}\]

\[=\left( 2x-1 \right)\left( 2x-1 \right)\]

∴ নির্ণেয় গ.সা.গু. = \left( 2x-1 \right)

(xii)

প্রথম সংখ্যামালা, {{x}^{3}}-3{{x}^{2}}-10x

\[=x\left\{ {{x}^{2}}-3x-10 \right\}\]

\[=x\left\{ {{x}^{2}}-\left( 5-2 \right)x-10 \right\}\]

\[=x\left\{ {{x}^{2}}-5x+2x-10 \right\}\]

\[=x\left\{ x\left( x-5 \right)+2\left( x-5 \right) \right\}\]

\[=x\left( x-5 \right)\left( x+2 \right)\]

দ্বিতীয় সংখ্যামালা,{{x}^{3}}+6{{x}^{2}}+8x

\[=x\left\{ {{x}^{2}}+6x+8 \right\}\]

\[=x\left\{ {{x}^{2}}+\left( 4+2 \right)x+8 \right\}\]

\[=x\left\{ {{x}^{2}}+4x+2x+8 \right\}\]

\[=x\left\{ x\left( x+4 \right)+2\left( x+4 \right) \right\}\]

\[=x\left( x+4 \right)\left( x+2 \right)\]

তৃতীয় সংখ্যামালা,{{x}^{4}}-5{{x}^{3}}-14{{x}^{2}}

\[={{x}^{2}}\left\{ {{x}^{2}}-5x-14 \right\}\]

\[={{x}^{2}}\left\{ {{x}^{2}}-\left( 7-2 \right)x-14 \right\}\]

\[={{x}^{2}}\left\{ {{x}^{2}}-7x+2x-14 \right\}\]

\[={{x}^{2}}\left\{ x\left( x-7 \right)+2\left( x-7 \right) \right\}\]

\[={{x}^{2}}\left( x-7 \right)\left( x+2 \right)\]

∴ নির্ণেয় গ.সা.গু. = x\left( x+2 \right)

(xiii)

প্রথম সংখ্যামালা, 6{{x}^{2}}-13xa+6{{a}^{2}}

\[=6{{x}^{2}}-\left( 9+4 \right)xa+6{{a}^{2}}\]

\[=6{{x}^{2}}-9xa-4xa+6{{a}^{2}}\]

\[=3x\left( 2x-3a \right)-2a\left( 2x-3a \right)\]

\[=\left( 2x-3a \right)\left( 3x-2a \right)\]

দ্বিতীয় সংখ্যামালা,6{{x}^{2}}+11xa-10{{a}^{2}}

\[=6{{x}^{2}}+\left( 15-4 \right)xa-10{{a}^{2}}\]

\[=6{{x}^{2}}+15xa-4xa-10{{a}^{2}}\]

\[=3x\left( 2x+5a \right)-2a\left( 2x+5a \right)\]

\[=\left( 2x+5a \right)\left( 3x-2a \right)\]

তৃতীয় সংখ্যামালা,6{{x}^{2}}+2xa-4{{a}^{2}}

\[=2\left\{ 3{{x}^{2}}+xa-2{{a}^{2}} \right\}\]

\[=2\left\{ 3{{x}^{2}}+\left( 3-2 \right)xa-2{{a}^{2}} \right\}\]

\[=2\left\{ 3{{x}^{2}}+3xa-2xa-2{{a}^{2}} \right\}\]

\[=2\left\{ 3x\left( x+a \right)-2a\left( x+a \right) \right\}\]

\[=2\left( x+a \right)\left( 3x-2a \right)\]

∴ নির্ণেয় গ.সা.গু. = \left( 3x-2a \right)

4. নীচের বীজগাণিতিক সংখ্যামালাগুলির ল.সা.গু. নির্ণয় করি –

\left( i \right){{p}^{2}}-{{q}^{2}},{{\left( p+q \right)}^{2}}          \left( ii \right)\left( {{x}^{2}}{{y}^{2}}-{{x}^{2}} \right),\left( x{{y}^{2}}-2xy+x \right)
\left( iii \right)\left( p+q \right)\left( p+r \right),\left( q+r \right)\left( r+p \right),\left( r+p \right)\left( p+q \right)          \left( iv \right)a{{b}^{4}}-8ab,{{a}^{2}}{{b}^{4}}+8{{a}^{2}}b,a{{b}^{4}}-4a{{b}^{2}}
\left( v \right){{x}^{4}}+{{x}^{2}}{{y}^{2}}+{{y}^{4}},{{x}^{3}}y+{{y}^{4}},{{\left( {{x}^{2}}-xy \right)}^{3}}          \left( vi \right){{p}^{2}}+2p,2{{p}^{4}}+3{{p}^{3}}-2{{p}^{2}},2{{p}^{3}}-3{{p}^{2}}-14p
\left( vii \right){{x}^{2}}-2xz+{{z}^{2}}-{{y}^{2}},{{x}^{2}}-{{y}^{2}}-{{z}^{2}}-2yz,xy+zx+{{y}^{2}}-{{z}^{2}}          \left( viii \right){{x}^{2}}-xy-2{{y}^{2}},2{{x}^{2}}-5xy+2{{y}^{2}},2{{x}^{2}}+xy-{{y}^{2}}
\left( ix \right)3{{x}^{2}}-15x+18,2{{x}^{2}}+2x-24,4{{x}^{2}}+36x+80     \left( x \right){{\left( {{a}^{2}}+2a \right)}^{2}},2{{a}^{3}}+3{{a}^{2}}-2a,2{{a}^{4}}-3{{a}^{3}}-14{{a}^{2}}
\left( xi \right)3{{a}^{2}}-5ab-12{{b}^{2}},{{a}^{5}}-27{{a}^{2}}{{b}^{3}},9{{a}^{2}}+24ab+16{{b}^{2}}

উত্তর-

(i)

প্রথম সংখ্যামালা,{{p}^{2}}-{{q}^{2}}=\left( p+q \right)\left( p-q \right)

দ্বিতীয় সংখ্যামালা,{{\left( p+q \right)}^{2}}=\left( p+q \right)\left( p+q \right)

∴ নির্ণেয় ল.সা.গু. =\left( p+q \right)\left( p+q \right)\left( p-q \right)={{\left( p+q \right)}^{2}}\left( p-q \right)

(ii)

প্রথম সংখ্যামালা,\left( {{x}^{2}}{{y}^{2}}-{{x}^{2}} \right)

\[={{x}^{2}}\left( {{y}^{2}}-1 \right)\]

\[=x\times x\left( y+1 \right)\left( y-1 \right)\]

দ্বিতীয় সংখ্যামালা,\left( x{{y}^{2}}-2xy+x \right)

\[=x\left( {{y}^{2}}-2y+1 \right)\]

\[=x{{\left( y-1 \right)}^{2}}\]

\[=x\left( y-1 \right)\left( y-1 \right)\]

∴ নির্ণেয় ল.সা.গু. =x\times x\left( y+1 \right)\left( y-1 \right)\left( y-1 \right)={{x}^{2}}\left( y+1 \right){{\left( y-1 \right)}^{2}}

(iii)

প্রথম সংখ্যামালা,\left( p+q \right)\left( p+r \right)

দ্বিতীয় সংখ্যামালা,\left( q+r \right)\left( r+p \right)=\left( q+r \right)\left( p+r \right)

তৃতীয় সংখ্যামালা,\left( r+p \right)\left( p+q \right)=\left( p+r \right)\left( p+q \right)

∴ নির্ণেয় ল.সা.গু. =\left( p+q \right)\left( p+r \right)\left( q+r \right)

(iv)

প্রথম সংখ্যামালা,a{{b}^{4}}-8ab

\[=ab\left( {{b}^{3}}-8 \right)\]

\[=ab\left\{ {{\left( b \right)}^{3}}-{{\left( 2 \right)}^{3}} \right\}\]

\[=ab\left( b-2 \right)\left( {{b}^{2}}+2b+4 \right)\]

দ্বিতীয় সংখ্যামালা,{{a}^{2}}{{b}^{4}}+8{{a}^{2}}b

\[={{a}^{2}}b\left( {{b}^{3}}+8 \right)\]

\[={{a}^{2}}b\left\{ {{\left( b \right)}^{3}}+{{\left( 2 \right)}^{3}} \right\}\]

\[={{a}^{2}}b\left( b+2 \right)\left( {{b}^{2}}-2b+4 \right)\]

তৃতীয় সংখ্যামালা,=a{{b}^{2}}\left( {{b}^{2}}-4 \right)

\[=a{{b}^{2}}\left\{ {{\left( b \right)}^{2}}-{{\left( 2 \right)}^{2}} \right\}\]

\[=a{{b}^{2}}\left( b+2 \right)\left( b-2 \right)\]

∴ নির্ণেয় ল.সা.গু. ={{a}^{2}}{{b}^{2}}\left( b-2 \right)\left( b+2 \right)\left( {{b}^{2}}-2b+4 \right)\left( {{b}^{2}}+2b+4 \right)

(v)

প্রথম সংখ্যামালা,{{x}^{4}}+{{x}^{2}}{{y}^{2}}+{{y}^{4}}

\[={{x}^{4}}+2{{x}^{2}}{{y}^{2}}+{{y}^{4}}-{{x}^{2}}{{y}^{2}}\]

\[={{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}-{{\left( xy \right)}^{2}}\]

\[=\left( {{x}^{2}}+{{y}^{2}}+xy \right)\left( {{x}^{2}}+{{y}^{2}}-xy \right)\]

দ্বিতীয় সংখ্যামালা,{{x}^{3}}y+{{y}^{4}}

\[=y\left( {{x}^{3}}+{{y}^{3}} \right)\]

\[=y\left( x+y \right)\left( {{x}^{2}}-xy+{{y}^{2}} \right)\]

তৃতীয় সংখ্যামালা,{{\left( {{x}^{2}}-xy \right)}^{3}}

\[={{\left\{ x\left( x-y \right) \right\}}^{3}}\]

\[={{x}^{3}}{{\left( x-y \right)}^{3}}\]

∴ নির্ণেয় ল.সা.গু. =\left( {{x}^{2}}+{{y}^{2}}+xy \right)\left( {{x}^{2}}+{{y}^{2}}-xy \right)y\left( x+y \right){{x}^{3}}{{\left( x-y \right)}^{3}}

\[={{x}^{3}}y\left( x+y \right){{\left( x-y \right)}^{3}}\left( {{x}^{2}}+{{y}^{2}}+xy \right)\left( {{x}^{2}}+{{y}^{2}}-xy \right)\]

(vi)

প্রথম সংখ্যামালা,{{p}^{2}}+2p=p\left( p+2 \right)

দ্বিতীয় সংখ্যামালা,2{{p}^{4}}+3{{p}^{3}}-2{{p}^{2}}

\[={{p}^{2}}\left( 2{{p}^{2}}+3p-2 \right)\]

\[={{p}^{2}}\left\{ 2{{p}^{2}}+\left( 4-1 \right)p-2 \right\}\]

\[={{p}^{2}}\left\{ 2{{p}^{2}}+4p-p-2 \right\}\]

\[={{p}^{2}}\left\{ 2p\left( p+2 \right)-1\left( p+2 \right) \right\}\]

\[={{p}^{2}}\left( p+2 \right)\left( 2p-1 \right)\]

তৃতীয় সংখ্যামালা,2{{p}^{3}}-3{{p}^{2}}-14p

\[=p\left\{ 2{{p}^{2}}-3p-14 \right\}\]

\[=p\left\{ 2{{p}^{2}}-\left( 7-4 \right)p-14 \right\}\]

\[=p\left\{ 2{{p}^{2}}-7p+4p-14 \right\}\]

\[=p\left\{ p\left( 2p-7 \right)+2\left( 2p-7 \right) \right\}\]

\[=p\left( 2p-7 \right)\left( p+2 \right)\]

∴ নির্ণেয় ল.সা.গু. ={{p}^{2}}\left( p+2 \right)\left( 2p-1 \right)\left( 2p-7 \right)

(vii)

প্রথম সংখ্যামালা,{{x}^{2}}-{{y}^{2}}+{{z}^{2}}-2xz

\[={{x}^{2}}-2xz+{{z}^{2}}-{{y}^{2}}\]

\[={{\left( x-z \right)}^{2}}-{{y}^{2}}\]

\[=\left( x-z+y \right)\left( x-z-y \right)\]

দ্বিতীয় সংখ্যামালা,{{x}^{2}}-{{y}^{2}}-{{z}^{2}}+2yz

\[={{x}^{2}}-\left\{ {{y}^{2}}+{{z}^{2}}-2yz \right\}\]

\[={{x}^{2}}-{{\left( y-z \right)}^{2}}\]

\[=\left( x+y-z \right)\left( x-y+z \right)\]

তৃতীয় সংখ্যামালা,xy+zx+{{y}^{2}}-{{z}^{2}}

\[=x\left( y+z \right)+\left( y+z \right)\left( y-z \right)\]

\[=\left( y+z \right)\left( x+y-z \right)\]

∴ নির্ণেয় ল.সা.গু. =\left( x-z+y \right)\left( x-z-y \right)\left( x+y-z \right)\left( x-y+z \right)\left( y+z \right)

(viii)

প্রথম সংখ্যামালা,{{x}^{2}}-xy-2{{y}^{2}}

\[={{x}^{2}}-\left( 2-1 \right)xy-2{{y}^{2}}\]

\[={{x}^{2}}-2xy+xy-2{{y}^{2}}\]

\[=x\left( x-2y \right)+y\left( x-2y \right)\]

\[=\left( x-2y \right)\left( x+y \right)\]

দ্বিতীয় সংখ্যামালা,2{{x}^{2}}-5xy+2{{y}^{2}}

\[=2{{x}^{2}}-\left( 4+1 \right)xy+2{{y}^{2}}\]

\[=2{{x}^{2}}-4xy-xy+2{{y}^{2}}\]

\[=2x\left( x-2y \right)-y\left( x-2y \right)\]

\[=\left( x-2y \right)\left( 2x-y \right)\]

তৃতীয় সংখ্যামালা,2{{x}^{2}}+xy-{{y}^{2}}

\[=2{{x}^{2}}+\left( 2-1 \right)xy-{{y}^{2}}\]

\[=2{{x}^{2}}+2xy-xy-{{y}^{2}}\]

\[=2x\left( x-y \right)-y\left( x-y \right)\]

\[=\left( x-y \right)\left( 2x-y \right)\]

∴ নির্ণেয় ল.সা.গু. =\left( x-2y \right)\left( x+y \right)\left( 2x-y \right)

(ix)

প্রথম সংখ্যামালা,3{{x}^{2}}-15x+18

\[=3\left( {{x}^{2}}-5x+6 \right)\]

\[=3{{x}^{2}}-9x-6x+18\]

\[=3\left\{ {{x}^{2}}-\left( 3+2 \right)x+6 \right\}\]

\[=3\left\{ {{x}^{2}}-3x-2x+6 \right\}\]

\[=3\left\{ x\left( x-3 \right)-2\left( x-3 \right) \right\}\]

\[=3\left( x-3 \right)\left( x-2 \right)\]

দ্বিতীয় সংখ্যামালা,2{{x}^{2}}+2x-24

\[=2\left\{ {{x}^{2}}+x-12 \right\}\]

\[=2\left\{ {{x}^{2}}+\left( 4-3 \right)x-12 \right\}\]

\[=2\left\{ {{x}^{2}}+4x-3x-12 \right\}\]

\[=2\left\{ x\left( x+4 \right)-3\left( x+4 \right) \right\}\]

\[=2\left( x+4 \right)\left( x-3 \right)\]

তৃতীয় সংখ্যামালা,4{{x}^{2}}+36x+80

\[=4\left\{ {{x}^{2}}+9x+20 \right\}\]

\[=4\left\{ {{x}^{2}}+\left( 5+4 \right)x+20 \right\}\]

\[=4\left\{ {{x}^{2}}+5x+4x+20 \right\}\]

\[=2\times 2\left\{ x\left( x+5 \right)+4\left( x+5 \right) \right\}\]

\[=2\times 2\left( x+5 \right)\left( x+4 \right)\]

∴ নির্ণেয় ল.সা.গু. =3\left( x-3 \right)\left( x-2 \right)\times 2\times 2\left( x+5 \right)\left( x+4 \right)

\[=12\left( x-3 \right)\left( x-2 \right)\left( x+4 \right)\left( x+5 \right)\]

(x)

প্রথম সংখ্যামালা,{{\left( {{a}^{2}}+2a \right)}^{2}}

\[={{\left\{ a\left( a+2 \right) \right\}}^{2}}\]

\[={{a}^{2}}{{\left( a+2 \right)}^{2}}\]

\[={{a}^{2}}\left( a+2 \right)\left( a+2 \right)\]

দ্বিতীয় সংখ্যামালা,2{{a}^{3}}+3{{a}^{2}}-2a

\[=a\left\{ 2{{a}^{2}}+3a-2 \right\}\]

\[=a\left\{ 2{{a}^{2}}+\left( 4-1 \right)a-2 \right\}\]

\[=a\left\{ 2{{a}^{2}}+4a-a-2 \right\}\]

\[=a\left\{ 2a\left( a+2 \right)-1\left( a+2 \right) \right\}\]

\[=a\left( a+2 \right)\left( 2a-1 \right)\]

তৃতীয় সংখ্যামালা,2{{a}^{4}}-3{{a}^{3}}-14{{a}^{2}}

\[={{a}^{2}}\left\{ 2{{a}^{2}}-3a-14 \right\}\]

\[={{a}^{2}}\left\{ 2{{a}^{2}}-\left( 7-4 \right)a-14 \right\}\]

\[={{a}^{2}}\left\{ 2{{a}^{2}}-7a+4a-14 \right\}\]

\[={{a}^{2}}\left\{ a\left( 2a-7 \right)+2\left( 2a-7 \right) \right\}\]

\[={{a}^{2}}\left( 2a-7 \right)\left( a+2 \right)\]

∴ নির্ণেয় ল.সা.গু. ={{a}^{2}}{{\left( a+2 \right)}^{2}}\left( 2a-1 \right)\left( 2a-7 \right)

(xi)

প্রথম সংখ্যামালা,3{{a}^{2}}-5ab-12{{b}^{2}}

\[=3{{a}^{2}}-\left( 9-4 \right)ab-12{{b}^{2}}\]

\[=3{{a}^{2}}-9ab+4ab-12{{b}^{2}}\]

\[=3a\left( a-3b \right)+4b\left( a-3b \right)\]

\[=\left( a-3b \right)\left( 3a+4b \right)\]

দ্বিতীয় সংখ্যামালা,{{a}^{5}}-27{{a}^{2}}{{b}^{3}}

\[={{a}^{2}}\left\{ {{a}^{3}}-27{{b}^{3}} \right\}\]

\[={{a}^{2}}\left\{ {{\left( a \right)}^{3}}-{{\left( 3b \right)}^{3}} \right\}\]

\[={{a}^{2}}\left( a-3b \right)\left( {{a}^{2}}+3ab+9{{b}^{2}} \right)\]

তৃতীয় সংখ্যামালা,9{{a}^{2}}+24ab+16{{b}^{2}}

\[={{\left( 3a \right)}^{2}}+2.3a.4b+{{\left( 4b \right)}^{2}}\]

\[={{\left( 3a+4b \right)}^{2}}\]

\[=\left( 3a+4b \right)\left( 3a+4b \right)\]

∴ নির্ণেয় ল.সা.গু. ={{a}^{2}}\left( a-3b \right){{\left( 3a+4b \right)}^{2}}\left( {{a}^{2}}+3ab+9{{b}^{2}} \right)

5. নীচের বীজগাণিতিক সংখ্যামালাগুলির গ.সা.গু. ও ল.সা.গু.  নির্ণয় করি –

\left( i \right){{x}^{3}}-8,{{x}^{2}}+3x-10,{{x}^{3}}+2{{x}^{2}}-8x          \left( ii \right)3{{y}^{2}}-15y+18,2{{y}^{2}}+2y-24,4{{y}^{2}}+36y+80
\left( iii \right){{a}^{3}}-4{{a}^{2}}+4a,{{a}^{2}}+a-6,{{a}^{3}}-8          \left( iv \right){{a}^{2}}+{{b}^{2}}-{{c}^{2}}+2ab,{{c}^{2}}+{{a}^{2}}-{{b}^{2}}+2ca,{{b}^{2}}+{{c}^{2}}-{{a}^{2}}+2bc
\left( v \right){{x}^{3}}-4x,4\left( {{x}^{2}}-5x+6 \right),\left( {{x}^{2}}-4x+4 \right)

উত্তর-

(i)

প্রথম সংখ্যামালা,{{x}^{3}}-8

\[={{\left( x \right)}^{3}}-{{\left( 2 \right)}^{3}}\]

\[=\left( x-2 \right)\left( {{x}^{2}}+2x+4 \right)\]

দ্বিতীয় সংখ্যামালা,{{x}^{2}}+3x-10

\[={{x}^{2}}+\left( 5-2 \right)x-10\]

\[={{x}^{2}}+5x-2x-10\]

\[=x\left( x+5 \right)-2\left( x+5 \right)\]

\[=\left( x+5 \right)\left( x-2 \right)\]

তৃতীয় সংখ্যামালা,{{x}^{3}}+2{{x}^{2}}-8x

\[=x\left\{ {{x}^{2}}+2x-8 \right\}\]

\[=x\left\{ {{x}^{2}}+\left( 4-2 \right)x-8 \right\}\]

\[=x\left\{ {{x}^{2}}+4x-2x-8 \right\}\]

\[=x\left\{ x\left( x+4 \right)-2\left( x+4 \right) \right\}\]

\[=x\left( x+4 \right)\left( x-2 \right)\]

∴ নির্ণেয় গ.সা.গু. =\left( x-2 \right)

∴ নির্ণেয় ল.সা.গু. =\left( x-2 \right)\left( {{x}^{2}}+2x+4 \right)\left( x+5 \right)x\left( x+4 \right)

\[=x\left( x-2 \right)\left( x+4 \right)\left( x+5 \right)\left( {{x}^{2}}+2x+4 \right)\]

(ii)

প্রথম সংখ্যামালা,3{{y}^{2}}-15y+18

\[=3\left\{ {{y}^{2}}-5y+6 \right\}\]

\[=3\left\{ {{y}^{2}}-\left( 3+2 \right)y+6 \right\}\]

\[=3\left\{ {{y}^{2}}-3y-2y+6 \right\}\]

\[=3\left\{ y\left( y-3 \right)-2\left( y-3 \right) \right\}\]

\[=3\left( y-3 \right)\left( y-2 \right)\]

দ্বিতীয় সংখ্যামালা,2{{y}^{2}}+2y-24

\[=2\left\{ {{y}^{2}}+y-12 \right\}\]

\[=2\left\{ {{y}^{2}}+\left( 4-3 \right)y-12 \right\}\]

\[=2\left\{ {{y}^{2}}+4y-3y-12 \right\}\]

\[=2\left\{ y\left( y+4 \right)-3\left( y+4 \right) \right\}\]

\[=2\left( y+4 \right)\left( y-3 \right)\]

তৃতীয় সংখ্যামালা,4{{y}^{2}}+36y+80

\[=4\left\{ {{y}^{2}}+9y+20 \right\}\]

\[=4\left\{ {{y}^{2}}+\left( 5+4 \right)y+20 \right\}\]

\[=4\left\{ {{y}^{2}}+5y+4y+20 \right\}\]

\[=4\left\{ y\left( y+5 \right)+4\left( y+5 \right) \right\}\]

\[=4\left( y+5 \right)\left( y+4 \right)\]

∴ নির্ণেয় গ.সা.গু. =1

∴ নির্ণেয় ল.সা.গু. =3\left( y-3 \right)\left( y-2 \right)\left( y+4 \right)4\left( y+5 \right)

\[=12\left( y-3 \right)\left( y-2 \right)\left( y+4 \right)\left( y+5 \right)\]

(iii)

প্রথম সংখ্যামালা,{{a}^{3}}-4{{a}^{2}}+4a

\[=a\left\{ {{a}^{2}}-4a+4 \right\}\]

\[=a\left\{ {{\left( a \right)}^{2}}-2.a.2+{{\left( 2 \right)}^{2}} \right\}\]

\[=a{{\left( a-2 \right)}^{2}}\]

\[=a\left( a-2 \right)\left( a-2 \right)\]

দ্বিতীয় সংখ্যামালা,{{a}^{2}}+a-6

\[={{a}^{2}}+\left( 3-2 \right)a-6\]

\[={{a}^{2}}+3a-2a-6\]

\[=a\left( a+3 \right)-2\left( a+3 \right)\]

\[=\left( a+3 \right)\left( a-2 \right)\]

তৃতীয় সংখ্যামালা,{{a}^{3}}-8

\[={{\left( a \right)}^{3}}-{{\left( 2 \right)}^{3}}\]

\[=\left( a-2 \right)\left( {{a}^{2}}+2a+4 \right)\]

∴ নির্ণেয় গ.সা.গু. =\left( a-2 \right)

∴ নির্ণেয় ল.সা.গু. =a\left( a-2 \right)\left( a-2 \right)\left( a+3 \right)\left( {{a}^{2}}+2a+4 \right)

\[=a{{\left( a-2 \right)}^{2}}\left( a+3 \right)\left( {{a}^{2}}+2a+4 \right)\]

(iv)

প্রথম সংখ্যামালা,{{a}^{2}}+{{b}^{2}}-{{c}^{2}}+2ab

\[={{a}^{2}}+2ab+{{b}^{2}}-{{c}^{2}}\]

\[={{\left( a+b \right)}^{2}}-{{c}^{2}}\]

\[=\left( a+b+c \right)\left( a+b-c \right)\]

দ্বিতীয় সংখ্যামালা,{{c}^{2}}+{{a}^{2}}-{{b}^{2}}+2ca

\[={{c}^{2}}+2ca+{{a}^{2}}-{{b}^{2}}\]

\[={{\left( c+a \right)}^{2}}-{{b}^{2}}\]

\[=\left( c+a+b \right)\left( c+a-b \right)\]

তৃতীয় সংখ্যামালা,{{b}^{2}}+{{c}^{2}}-{{a}^{2}}+2bc

\[={{b}^{2}}+2bc+{{c}^{2}}-{{a}^{2}}\]

\[={{\left( b+c \right)}^{2}}-{{a}^{2}}\]

\[=\left( b+c+a \right)\left( b+c-a \right)\]

∴ নির্ণেয় গ.সা.গু. =\left( a+b+c \right)

∴ নির্ণেয় ল.সা.গু. =\left( a+b+c \right)\left( a+b-c \right)\left( c+a-b \right)\left( b+c-a \right)

(v)

প্রথম সংখ্যামালা,{{x}^{3}}-4x

\[=x\left\{ {{x}^{2}}-4 \right\}\]

\[=x\left\{ {{\left( x \right)}^{2}}-{{\left( 2 \right)}^{2}} \right\}\]

\[=x\left( x+2 \right)\left( x-2 \right)\]

দ্বিতীয় সংখ্যামালা,4\left( {{x}^{2}}-5x+6 \right)

\[=4\left\{ {{x}^{2}}-\left( 3+2 \right)x+6 \right\}\]

\[=4\left\{ {{x}^{2}}-3x-2x+6 \right\}\]

\[=4\left\{ x\left( x-3 \right)-2\left( x-3 \right) \right\}\]

\[=4\left( x-3 \right)\left( x-2 \right)\]

তৃতীয় সংখ্যামালা,\left( {{x}^{2}}-4x+4 \right)

\[={{\left( x \right)}^{2}}-2.x.2+{{\left( 2 \right)}^{2}}\]

\[={{\left( x-2 \right)}^{2}}\]

\[=\left( x-2 \right)\left( x-2 \right)\]

∴ নির্ণেয় গ.সা.গু. =\left( x-2 \right)

∴ নির্ণেয় ল.সা.গু. =x\left( x+2 \right)\left( x-2 \right)4\left( x-3 \right)\left( x-2 \right)

\[=4x{{\left( x-2 \right)}^{2}}\left( x+2 \right)\left( x-3 \right)\]

;