Class 8 Chapter 13 সংখ্যামালার উৎপাদকে বিশ্লেষণ

অষ্টম শ্রেণি – অধ্যায় ১৩ : বীজগাণিতিক সংখ্যামালার উৎপাদকে বিশ্লেষণ সম্পূর্ণ সমাধান

কষে দেখি – 13.1

 

1. নিচের বীজগাণিতিক সংখ্যামালাগুলি  {{x}^{2}}+\left(p+q\right)x+pq=\left( x+p\right)\left( x+q\right) অভেদের সাথে তুলনা করে p ও q এর মান খুঁজে লিখি ও উৎপাদকে বিশ্লেষণ করি।

বীজগাণিতিক সংখ্যামালাp ও q – এর মানউৎপাদকে বিশ্লেষণ
{{x}^{2}}-8x+15p=-5,q=-3\left( x-5 \right)\left( x-3 \right)
{{x}^{2}}-40x-129  
{{m}^{2}}+19m+60  
{{x}^{2}}-x-6  
{{\left( a+b \right)}^{2}}-4\left( a+b \right)-12  
{{\left( x-y \right)}^{2}}-x+y-2  

উত্তরঃ-

বীজগাণিতিক সংখ্যামালাp ও q – এর মানউৎপাদকে বিশ্লেষণ
{{x}^{2}}-8x+15p=-5,q=-3\left( x-5 \right)\left( x-3 \right)
{{x}^{2}}-40x-129p=3,q=-43\left( x+3 \right)\left( x-43 \right)
{{m}^{2}}+19m+60p=15,q=4\left( m+15 \right)\left( m+4 \right)
{{x}^{2}}-x-6p=-3,q=2\left( x-3 \right)\left( x+2 \right)
{{\left( a+b \right)}^{2}}-4\left( a+b \right)-12p=-6,q=2\left( a+b-6 \right)\left( a+b+2 \right)
{{\left( x-y \right)}^{2}}-x+y-2p=-2,q=1\left( x-y-2 \right)\left( x-y+1 \right)
 

2. উৎপাদকে বিশ্লেষণ করি-

\left( i \right){{\left( a+b \right)}^{2}}-5\left( a+b \right)-6          (ii){{\left( {{x}^{2}}-2x \right)}^{2}}+5\left( {{x}^{2}}-2x \right)-36
(iii){{\left( {{p}^{2}}-3{{q}^{2}} \right)}^{2}}-16\left( {{p}^{2}}-3{{q}^{2}} \right)+63          (iv){{a}^{4}}+4{{a}^{2}}-5
\left( v \right){{x}^{2}}{{y}^{2}}+23xy-420          \left( vi \right){{x}^{4}}-7{{x}^{2}}+12
\left( vii \right){{a}^{2}}+ab-12{{b}^{2}}          \left( viii \right){{p}^{2}}+31pq+108{{q}^{2}}
\left( ix \right){{a}^{6}}+3{{a}^{3}}{{b}^{3}}-40{{b}^{6}}          \left( x \right)\left( x+1 \right)\left( x+3 \right)\left( x-4 \right)\left( x-6 \right)+24
\left( xi \right)\left( x+1 \right)\left( x+9 \right){{\left( x+5 \right)}^{2}}+63          \left( xii \right)x\left( x+2 \right)\left( x+6 \right)\left( x+9 \right)+56
\left( xiii \right){{x}^{2}}-2ax+\left( a+b \right)\left( a-b \right)          \left( xiv \right){{x}^{2}}-bx-\left( a+3b \right)\left( a+2b \right)
\left( xv \right){{\left( a+b \right)}^{2}}-5a-5b+6          \left( xvi \right){{x}^{2}}+4abx-{{\left( {{a}^{2}}-{{b}^{2}} \right)}^{2}}
\left( xvii \right){{x}^{2}}-\left( a+\frac{1}{a} \right)x+1          \left( xviii \right){{x}^{6}}{{y}^{6}}-9{{x}^{3}}{{y}^{3}}+8

উত্তরঃ-

\[(i){{\left( a+b \right)}^{2}}-5a-5b+6\]

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

\left( a+b \right)=x ধরে পাই,

\[{{x}^{2}}-5x+6\]

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

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

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

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

x=\left( a+b \right) বসিয়ে পাই,

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

\[(ii){{\left( {{x}^{2}}-2x \right)}^{2}}+5\left( {{x}^{2}}-2x \right)-36\]

ধরি, {{x}^{2}}-2x=a

\[\therefore {{a}^{2}}+5a-36\]

\[={{a}^{2}}+\left( 9-4 \right)a-36\]

\[={{a}^{2}}+9a-4a-36\]

\[=a\left( a+9 \right)-4\left( a+9 \right)\]

\[=\left( a+9 \right)\left( a-4 \right)\]

a={{x}^{2}}-2x বসিয়ে পাই,

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

\[(iii){{\left( {{p}^{2}}-3{{q}^{2}} \right)}^{2}}-16\left( {{p}^{2}}-3{{q}^{2}} \right)+63\]

ধরি,{{p}^{2}}-3{{q}^{2}}=x

\[{{x}^{2}}-16x+63\]

\[={{x}^{2}}-\left( 9+7 \right)x+63\]

\[={{x}^{2}}-9x-7x+63\]

\[=x\left( x-9 \right)-7\left( x-9 \right)\]

\[=\left( x-9 \right)\left( x-7 \right)\]

x={{p}^{2}}-3{{q}^{2}} বসিয়ে পাই,

\[\left( {{p}^{2}}-3{{q}^{2}}-9 \right)\left( {{p}^{2}}-3{{q}^{2}}-7 \right)\]

\[(iv){{a}^{4}}+4{{a}^{2}}-5\]

ধরি, {{a}^{2}}=x

\[\therefore {{x}^{2}}+4x-5\]

\[={{x}^{2}}+\left( 5-1 \right)x-5\]

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

\[=x\left( x+5 \right)-1\left( x+5 \right)\]

\[=\left( x+5 \right)\left( x+1 \right)\]

x={{a}^{2}} বসিয়ে পাই,

\[\left( {{a}^{2}}+5 \right)\left( {{a}^{2}}+1 \right)\]

\[\left( v \right){{x}^{2}}{{y}^{2}}+23xy-420\]

\[={{x}^{2}}{{y}^{2}}+\left( 35-12 \right)xy-420\]

\[={{x}^{2}}{{y}^{2}}+35xy-12xy-420\]

\[=xy\left( xy+35 \right)-12\left( xy+35 \right)\]

\[=\left( xy+35 \right)\left( xy-12 \right)\]

\[\left( vi \right){{x}^{4}}-7{{x}^{2}}+12\]

ধরি, {{x}^{2}}=a

\[\therefore {{a}^{2}}-7a+12\]

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

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

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

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

a={{x}^{2}} বসিয়ে পাই,

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

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

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

\[\left( vii \right){{a}^{2}}+ab-12{{b}^{2}}\]

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

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

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

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

\[\left( viii \right){{p}^{2}}+31pq+108{{q}^{2}}\]

\[={{p}^{2}}+\left( 27+4 \right)pq+108{{q}^{2}}\]

\[={{p}^{2}}+27pq+4pq+108{{q}^{2}}\]

\[=p\left( p+27q \right)+4q\left( p+27q \right)\]

\[=\left( p+27q \right)\left( p+4q \right)\]

\[\left( ix \right){{a}^{6}}+3{{a}^{3}}{{b}^{3}}-40{{b}^{6}}\]

ধরি, {{a}^{3}}=x,{{b}^{3}}=y

\[\therefore {{x}^{2}}+3xy-40{{y}^{2}}\]

\[={{x}^{2}}+\left( 8-5 \right)xy-40{{y}^{2}}\]

\[={{x}^{2}}+8xy-5xy-40{{y}^{2}}\]

\[=x\left( x+8y \right)-5y\left( x+8y \right)\]

\[=\left( x+8y \right)\left( x-5y \right)\]

x={{a}^{3}},y={{b}^{3}} বসিয়ে পাই,

\[=\left( {{a}^{3}}+8{{b}^{3}} \right)\left( {{a}^{3}}-5{{b}^{3}} \right)\]

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

\[=\left( a+2b \right)\left( {{a}^{2}}-2ab+4{{b}^{2}} \right)\left( {{a}^{3}}-5{{b}^{3}} \right)\]

\[\left( x \right)\left( x+1 \right)\left( x+3 \right)\left( x-4 \right)\left( x-6 \right)+24\]

\[=\left\{ \left( x+1 \right)\left( x-4 \right) \right\}\left\{ \left( x+3 \right)\left( x-6 \right) \right\}+24\]

\[=\left( {{x}^{2}}-4x+x-4 \right)\left( {{x}^{2}}-6x+3x-18 \right)+24\]

\[=\left( {{x}^{2}}-3x-4 \right)\left( {{x}^{2}}-3x-18 \right)+24\]

ধরি, {{x}^{2}}-3x=a

\[\left( a-4 \right)\left( a-18 \right)+24\]

\[={{a}^{2}}-18a-4a+72+24\]

\[={{a}^{2}}-22a+96\]

\[={{a}^{2}}-\left( 16+6 \right)a+96\]

\[={{a}^{2}}-16a-6a+96\]

\[=a\left( a-16 \right)-6\left( a-16 \right)\]

\[=\left( a-16 \right)\left( a-6 \right)\]

a={{x}^{2}}-3x বসিয়ে পাই,

\[\left( {{x}^{2}}-3x-16 \right)\left( {{x}^{2}}-3x-6 \right)\]

\[\left( xi \right)\left( x+1 \right)\left( x+9 \right){{\left( x+5 \right)}^{2}}+63\]

\[=\left( {{x}^{2}}+9x+x+9 \right)\left( {{x}^{2}}+10x+25 \right)+63\]

\[=\left( {{x}^{2}}+10x+9 \right)\left( {{x}^{2}}+10x+25 \right)+63\]

ধরি, {{x}^{2}}+10x=a

\[\therefore \left( a+9 \right)\left( a+25 \right)+63\]

\[={{a}^{2}}+25a+9a+225+63\]

\[={{a}^{2}}+34a+288\]

\[={{a}^{2}}+\left( 18+16 \right)a+288\]

\[={{a}^{2}}+18a+16a+288\]

\[=a\left( a+18 \right)+16\left( a+18 \right)\]

\[=\left( a+18 \right)\left( a+16 \right)\]

a={{x}^{2}}+10x বসিয়ে পাই,

\[=\left( {{x}^{2}}+10x+18 \right)\left( {{x}^{2}}+10x+16 \right)\]

\[=\left( {{x}^{2}}+10x+18 \right)\left\{ {{x}^{2}}+\left( 8+2 \right)x+16 \right\}\]

\[=\left( {{x}^{2}}+10x+18 \right)\left\{ {{x}^{2}}+8x+2x+16 \right\}\]

\[=\left( {{x}^{2}}+10x+18 \right)\left\{ x\left( x+8 \right)+2\left( x+8 \right) \right\}\]

\[=\left( {{x}^{2}}+10x+18 \right)\left( x+8 \right)\left( x+2 \right)\]

\[\left( xii \right)x\left( x+2 \right)\left( x+6 \right)\left( x+9 \right)+56\]

\[=\left\{ x\left( x+9 \right) \right\}\left\{ \left( x+3 \right)\left( x+6 \right) \right\}+56\]

\[=\left( {{x}^{2}}+9x \right)\left( {{x}^{2}}+6x+3x+18 \right)+56\]

\[=\left( {{x}^{2}}+9x \right)\left( {{x}^{2}}+9x+18 \right)+56\]

ধরি,{{x}^{2}}+9x=a

\[\therefore a\left( a+18 \right)+56\]

\[={{a}^{2}}+18a+56\]

\[={{a}^{2}}+\left( 14+4 \right)a+56\]

\[={{a}^{2}}+14a+4a+56\]

\[=a\left( a+14 \right)+4\left( a+14 \right)\]

\[=\left( a+14 \right)\left( a+4 \right)\]

a={{x}^{2}}+9x বসিয়ে পাই,

\[\left( {{x}^{2}}+9x+14 \right)\left( {{x}^{2}}+9x+4 \right)\]

\[=\left\{ {{x}^{2}}+\left( 7+2 \right)x+14 \right\}\left( {{x}^{2}}+9x+4 \right)\]

\[=\left\{ {{x}^{2}}+7x+2x+14 \right\}\left( {{x}^{2}}+9x+4 \right)\]

\[=\left\{ x\left( x+7 \right)+2\left( x+7 \right) \right\}\left( {{x}^{2}}+9x+4 \right)\]

\[=\left( x+7 \right)\left( x+2 \right)\left( {{x}^{2}}+9x+4 \right)\]

\[\left( xiii \right){{x}^{2}}-2ax+\left( a+b \right)\left( a-b \right)\]

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

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

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

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

\[\left( xiv \right){{x}^{2}}-bx-\left( a+3b \right)\left( a+2b \right)\]

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

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

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

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

\[\left( xv \right){{\left( a+b \right)}^{2}}-5a-5b+6\]

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

ধরি, \left( a+b \right)=x

\[\therefore {{x}^{2}}-5x+6\]

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

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

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

x=\left( a+b \right) বসিয়ে পাই,

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

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

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

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

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

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

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

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

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

\[\left( xvii \right){{x}^{2}}-\left( a+\frac{1}{a} \right)x+1\]

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

\[={{x}^{2}}-ax-\frac{x}{a}+a\times \frac{1}{a}\]

\[=x\left( x-a \right)-\frac{1}{a}\left( x-a \right)\]

\[=\left( x-a \right)\left( x-\frac{1}{a} \right)\]

\[\left( xviii \right){{x}^{6}}{{y}^{6}}-9{{x}^{3}}{{y}^{3}}+8\]

ধরি,{{x}^{3}}{{y}^{3}}=a

\[\therefore {{a}^{2}}-9a+8\]

\[={{a}^{2}}-\left( 8+1 \right)a+8\]

\[={{a}^{2}}-8a-a+8\]

\[=a\left( a-8 \right)-1\left( a-8 \right)\]

\[=\left( a-8 \right)\left( a-1 \right)\]

a={{x}^{3}}{{y}^{3}} বসিয়ে পাই,

\[\left( {{x}^{3}}{{y}^{3}}-8 \right)\left( {{x}^{3}}{{y}^{3}}-1 \right)\]

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

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

কষে দেখি – 13.2

 

1. উৎপাদকে বিশ্লেষণ করি-

\left( i \right)2{{a}^{2}}+5a+2          \left( ii \right)3{{x}^{2}}+14x+8
\left( iii \right)2{{m}^{2}}+7m+6          \left( iv \right)6{{x}^{2}}-x-15
\left( v \right)9{{r}^{2}}+r-8          \left( vi \right)6{{m}^{2}}-11mn-10{{n}^{2}}
\left( vii \right)7{{x}^{2}}+48xy-7{{y}^{2}}          \left( viii \right)12+x-6{{x}^{2}}
\left( ix \right)6+5a-6{{a}^{2}}          \left( x \right)6{{x}^{2}}-13x+6
\left( xi \right)99{{a}^{2}}-202ab+99{{b}^{2}}          \left( xii \right)2{{a}^{6}}-13{{a}^{3}}-24
\left( xiii \right)8{{a}^{4}}+2{{a}^{2}}-45          \left( xiv \right)6{{\left( x-y \right)}^{2}}-x+y-15
\left( xv \right)3{{\left( a+b \right)}^{2}}-2a-2b-8          \left( xvi \right)6{{\left( a+b \right)}^{2}}+5\left( {{a}^{2}}-{{b}^{2}} \right)-6{{\left( a-b \right)}^{2}}

উত্তরঃ-

\[\left( i \right)2{{a}^{2}}+5a+2\]

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

\[=2{{a}^{2}}+4a+a+2\]

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

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

\[\left( ii \right)3{{x}^{2}}+14x+8\]

\[=3{{x}^{2}}+\left( 12+2 \right)x+8\]

\[=3{{x}^{2}}+12x+2x+8\]

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

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

\[\left( iii \right)2{{m}^{2}}+7m+6\]

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

\[=2{{m}^{2}}+4m+3m+6\]

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

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

\[\left( iv \right)6{{x}^{2}}-x-15\]

\[=6{{x}^{2}}-\left( 10-9 \right)x-15\]

\[=6{{x}^{2}}-10x+9x-15\]

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

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

\[\left( v \right)9{{r}^{2}}+r-8\]

\[=9{{r}^{2}}+\left( 9-8 \right)r-8\]

\[=9{{r}^{2}}+9r-8r-8\]

\[=9r\left( r+1 \right)-8\left( r+1 \right)\]

\[=\left( r+1 \right)\left( 9r-8 \right)\]

\[\left( vi \right)6{{m}^{2}}-11mn-10{{n}^{2}}\]

\[=6{{m}^{2}}-\left( 15-4 \right)mn-10{{n}^{2}}\]

\[=6{{m}^{2}}-15mn+4mn-10{{n}^{2}}\]

\[=3m\left( 2m-5n \right)+2n\left( 2m-5n \right)\]

\[=\left( 2m-5n \right)\left( 3m+2n \right)\]

\[\left( vii \right)7{{x}^{2}}+48xy-7{{y}^{2}}\]

\[=7{{x}^{2}}+\left( 49-1 \right)xy-7{{y}^{2}}\]

\[=7{{x}^{2}}+49xy-xy-7{{y}^{2}}\]

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

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

\[\left( viii \right)12+x-6{{x}^{2}}\]

\[=12+\left( 9-8 \right)x-6{{x}^{2}}\]

\[=12+9x-8x-6{{x}^{2}}\]

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

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

\[\left( ix \right)6+5a-6{{a}^{2}}\]

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

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

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

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

\[\left( x \right)6{{x}^{2}}-13x+6\]

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

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

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

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

\[\left( xi \right)99{{a}^{2}}-202ab+99{{b}^{2}}\]

\[=99{{a}^{2}}-\left( 121+81 \right)ab+99{{b}^{2}}\]

\[=99{{a}^{2}}-121ab-81ab+99{{b}^{2}}\]

\[=11a\left( 9a-11b \right)-9b\left( 9a-11b \right)\]

\[=\left( 9a-11b \right)\left( 11a-9b \right)\]

\[\left( xii \right)2{{a}^{6}}-13{{a}^{3}}-24\]

\[=2{{a}^{6}}-\left( 16-3 \right){{a}^{3}}-24\]

\[=2{{a}^{6}}-16{{a}^{3}}+3{{a}^{3}}-24\]

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

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

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

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

\[\left( xiii \right)8{{a}^{4}}+2{{a}^{2}}-45\]

\[=8{{a}^{4}}+\left( 20-18 \right){{a}^{2}}-45\]

\[=8{{a}^{4}}+20{{a}^{2}}-18{{a}^{2}}-45\]

\[=4{{a}^{2}}\left( 2{{a}^{2}}+5 \right)-9\left( 2{{a}^{2}}+5 \right)\]

\[=\left( 2{{a}^{2}}+5 \right)\left( 4{{a}^{2}}-9 \right)\]

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

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

\[\left( xiv \right)6{{\left( x-y \right)}^{2}}-x+y-15\]

\[=6{{\left( x-y \right)}^{2}}-\left( x-y \right)-15\]

ধরি, x-y=a

\[\therefore 6{{a}^{2}}-a-15\]

\[=6{{a}^{2}}-\left( 10-9 \right)a-15\]

\[=6{{a}^{2}}-10a+9a-15\]

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

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

a=x-y বসিয়ে পাই,

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

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

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

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

ধরি,a+b=x

\[\therefore 3{{x}^{2}}-2x-8\]

\[=3{{x}^{2}}-\left( 6-4 \right)x-8\]

\[=3{{x}^{2}}-6x+4x-8\]

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

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

x=a+b বসিয়ে পাই,

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

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

\[\left( xvi \right)6{{\left( a+b \right)}^{2}}+5\left( {{a}^{2}}-{{b}^{2}} \right)-6{{\left( a-b \right)}^{2}}\]

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

ধরি,a+b=x,a-b=y

\[\therefore 6{{x}^{2}}+5xy-6{{y}^{2}}\]

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

\[=6{{x}^{2}}+9xy-4xy-6{{y}^{2}}\]

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

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

x=a+b,y=a-b বসিয়ে পাই,

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

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

\[=\left( 5a-b \right)\left( a+5b \right)\]

 

2. নিচের বীজগাণিতিক সংখ্যামালাগুলি দুটি বর্গের অন্তররুপে প্রকাশ করে উৎপাদকে বিশ্লেষণ করি- 

\left( i \right){{x}^{2}}-2x-3          \left( ii \right){{x}^{2}}+5x+6
\left( iii \right)3{{x}^{2}}-7x-6          \left( iv \right)3{{a}^{2}}-2a-5

উত্তরঃ-

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

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

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

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

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

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

\[\left( ii \right){{x}^{2}}+5x+6\]

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

\[={{\left( x+\frac{5}{2} \right)}^{2}}-\left( \frac{25}{4}-6 \right)\]

\[={{\left( x+\frac{5}{2} \right)}^{2}}-\left( \frac{25-24}{4} \right)\]

\[={{\left( x+\frac{5}{2} \right)}^{2}}-\frac{1}{4}\]

\[={{\left( x+\frac{5}{2} \right)}^{2}}-{{\left( \frac{1}{2} \right)}^{2}}\]

\[=\left( x+\frac{5}{2}+\frac{1}{2} \right)\left( x+\frac{5}{2}-\frac{1}{2} \right)\]

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

\[\left( iii \right)3{{x}^{2}}-7x-6\]

\[=3\left( {{x}^{2}}-\frac{7}{3}x-2 \right)\]

\[=3\left\{ {{\left( x \right)}^{2}}-2.x.\frac{7}{6}+{{\left( \frac{7}{6} \right)}^{2}}-{{\left( \frac{7}{6} \right)}^{2}}-2 \right\}\]

\[=3\left\{ {{\left( x-\frac{7}{6} \right)}^{2}}-\left( \frac{49}{36}+2 \right) \right\}\]

\[=3\left\{ {{\left( x-\frac{7}{6} \right)}^{2}}-\left( \frac{49+72}{36} \right) \right\}\]

\[=3\left\{ {{\left( x-\frac{7}{6} \right)}^{2}}-\frac{121}{36} \right\}\]

\[=3\left\{ {{\left( x-\frac{7}{6} \right)}^{2}}-{{\left( \frac{11}{6} \right)}^{2}} \right\}\]

\[=3\left( x-\frac{7}{6}+\frac{11}{6} \right)\left( x-\frac{7}{6}-\frac{11}{6} \right)\]

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

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

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

\[\left( iv \right)3{{a}^{2}}-2a-5\]

\[=3\left( {{a}^{2}}-\frac{2}{3}a-\frac{5}{3} \right)\]

\[=3\left\{ {{\left( a \right)}^{2}}-2.a.\frac{1}{3}+{{\left( \frac{1}{3} \right)}^{2}}-{{\left( \frac{1}{3} \right)}^{2}}-\frac{5}{3} \right\}\]

\[=3\left\{ {{\left( a-\frac{1}{3} \right)}^{2}}-\left( \frac{1}{9}+\frac{5}{3} \right) \right\}\]

\[=3\left\{ {{\left( a-\frac{1}{3} \right)}^{2}}-\left( \frac{1+15}{9} \right) \right\}\]

\[=3\left\{ {{\left( a-\frac{1}{3} \right)}^{2}}-\left( \frac{16}{9} \right) \right\}\]

\[=3\left\{ {{\left( a-\frac{1}{3} \right)}^{2}}-{{\left( \frac{4}{3} \right)}^{2}} \right\}\]

\[=3\left( a-\frac{1}{3}+\frac{4}{3} \right)\left( a-\frac{1}{3}-\frac{4}{3} \right)\]

\[=3\left( a+1 \right)\left( a-\frac{5}{3} \right)\]

 

3. উৎপাদকে বিশ্লেষণ করি-

\left( i \right)a{{x}^{2}}+\left( {{a}^{2}}+1 \right)x+a          \left( ii \right){{x}^{2}}+2ax+\left( a+b \right)\left( a-b \right)
\left( iii \right)a{{x}^{2}}-\left( {{a}^{2}}+1 \right)x+a          \left( iv \right)a{{x}^{2}}+\left( {{a}^{2}}-1 \right)x-a
\left( v \right)a{{x}^{2}}-\left( {{a}^{2}}-2 \right)x-2a          \left( vi \right){{a}^{2}}+1-\frac{6}{{{a}^{2}}}

উত্তরঃ-

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

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

\[=ax\left( x+a \right)+1\left( x+a \right)\]

\[=\left( x+a \right)\left( ax+1 \right)\]

\[\left( ii \right){{x}^{2}}+2ax+\left( a+b \right)\left( a-b \right)\]

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

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

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

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

\[\left( iii \right)a{{x}^{2}}-\left( {{a}^{2}}+1 \right)x+a\]

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

\[=ax\left( x-a \right)-1\left( x-a \right)\]

\[=\left( x-a \right)\left( ax-1 \right)\]

\[\left( iv \right)a{{x}^{2}}+\left( {{a}^{2}}-1 \right)x-a\]

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

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

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

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

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

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

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

\[\left( vi \right){{a}^{2}}+1-\frac{6}{{{a}^{2}}}\]

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

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

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

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