অষ্টম শ্রেণি – অধ্যায় ১৫ : বীজগাণিতিক সংখ্যামালার সরলীকরণ সম্পূর্ণ সমাধান

কষে দেখি – 15

1. নীচের সম্পর্কগুলি দেখি ও কোনটি সত্য ও কোনটি মিথ্যা লিখি।

\left( i \right)\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}          \left( ii \right)\frac{a}{x+y}=\frac{a}{x}+\frac{a}{y}
\left( iii \right)\frac{x-y}{a-b}=\frac{y-x}{b-a}          \left( iv \right)\frac{1}{x}+\frac{1}{y}=\frac{1}{x+y}

উত্তর-

\left( i \right)\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}
ডানপক্ষ,
\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}
∴ বামপক্ষ = ডানপক্ষ।
সুতরাং, \frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}সম্পর্কটি সত্য।

\left( ii \right)\frac{a}{x+y}=\frac{a}{x}+\frac{a}{y}
ডানপক্ষ,
\frac{a}{x}+\frac{a}{y}=\frac{ay+ax}{xy}
∴ বামপক্ষ ≠ ডানপক্ষ, সম্পর্কটি মিথ্যা।

\left( iii \right)\frac{x-y}{a-b}=\frac{y-x}{b-a}
বামপক্ষ,
\frac{x-y}{a-b}=\frac{-\left( y-x \right)}{-\left( b-a \right)}=\frac{y-x}{b-a}
∴ বামপক্ষ = ডানপক্ষ , সম্পর্কটি সত্য।

\left( iv \right)\frac{1}{x}+\frac{1}{y}=\frac{1}{x+y}
বামপক্ষ,
\frac{1}{x}+\frac{1}{y}=\frac{y+x}{xy}
∴ বামপক্ষ ≠ ডানপক্ষ, সম্পর্কটি মিথ্যা।

2. নীচের বীজগাণিতিক ভগ্নাংশগুলি লঘিষ্ঠ আকারে প্রকাশ করি।

\left( i \right)\frac{63{{a}^{3}}{{b}^{4}}}{77{{b}^{5}}}          \left( ii \right)\frac{18{{a}^{4}}{{b}^{5}}{{c}^{2}}}{21{{a}^{7}}{{b}^{2}}}
\left( iii \right)\frac{{{x}^{2}}-3x+2}{{{x}^{2}}-1}          \left( iv \right)\frac{a+1}{a-2}\times \frac{{{a}^{2}}-a-2}{{{a}^{2}}+a}
\left( v \right)\frac{{{p}^{3}}+{{q}^{3}}}{{{p}^{2}}-{{q}^{2}}}\div \frac{p+q}{p-q}          \left( vi \right)\frac{{{x}^{2}}-x-6}{{{x}^{2}}+4x-5}\times \frac{{{x}^{2}}+6x+5}{{{x}^{2}}-4x+3}
\left( vii \right)\frac{{{a}^{2}}-ab+{{b}^{2}}}{{{a}^{2}}+ab}\div \frac{{{a}^{3}}+{{b}^{3}}}{{{a}^{2}}-{{b}^{2}}}

উত্তর-

\[\left( i \right)\frac{63{{a}^{3}}{{b}^{4}}}{77{{b}^{5}}}=\frac{9{{a}^{3}}}{11b}\]

\[\left( ii \right)\frac{18{{a}^{4}}{{b}^{5}}{{c}^{2}}}{21{{a}^{7}}{{b}^{2}}}=\frac{6{{b}^{3}}{{c}^{2}}}{7{{a}^{3}}}\]

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

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

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

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

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

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

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

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

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

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

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

\[=\frac{a+1}{a}\]

\[\left( v \right)\frac{{{p}^{3}}+{{q}^{3}}}{{{p}^{2}}-{{q}^{2}}}\div \frac{p+q}{p-q}\]

\[=\frac{\left( p+q \right)\left( {{p}^{2}}-pq+{{q}^{2}} \right)}{\left( p+q \right)\left( p-q \right)}\times \frac{p-q}{p+q}\]

\[=\frac{\left( {{p}^{2}}-pq+{{q}^{2}} \right)}{\left( p+q \right)}\]

\[\left( vi \right)\frac{{{x}^{2}}-x-6}{{{x}^{2}}+4x-5}\times \frac{{{x}^{2}}+6x+5}{{{x}^{2}}-4x+3}\]

\[=\frac{{{x}^{2}}-\left( 3-2 \right)x-6}{{{x}^{2}}+\left( 5-1 \right)x-5}\times \frac{{{x}^{2}}+\left( 5+1 \right)x+5}{{{x}^{2}}-\left( 3+1 \right)x+3}\]

\[=\frac{{{x}^{2}}-3x+2x-6}{{{x}^{2}}+5x-x-5}\times \frac{{{x}^{2}}+5x+x+5}{{{x}^{2}}-3x-x+3}\]

\[=\frac{x\left( x-3 \right)+2\left( x-3 \right)}{x\left( x+5 \right)-1\left( x+5 \right)}\times \frac{x\left( x+5 \right)+1\left( x+5 \right)}{x\left( x-3 \right)-1\left( x-3 \right)}\]

\[=\frac{\left( x-3 \right)\left( x+2 \right)}{\left( x+5 \right)\left( x-1 \right)}\times \frac{\left( x+5 \right)\left( x+1 \right)}{\left( x-3 \right)\left( x-1 \right)}\]

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

\[=\frac{{{x}^{2}}+3x+2}{{{x}^{2}}-2x+1}\]

\[\left( vii \right)\frac{{{a}^{2}}-ab+{{b}^{2}}}{{{a}^{2}}+ab}\div \frac{{{a}^{3}}+{{b}^{3}}}{{{a}^{2}}-{{b}^{2}}}\]

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

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

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

3. নীচের বীজগাণিতিক ভগ্নাংশগুলি সরলতম আকারে প্রকাশ করি।

\left( i \right)\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}          \left( ii \right)\frac{a-b-c}{a}+\frac{a+b+c}{a}
\left( iii \right)\frac{{{x}^{2}}+{{a}^{2}}}{ab}+\frac{x-a}{ax}-\frac{{{x}^{3}}}{b}          \left( iv \right)\frac{2{{a}^{2}}b}{3{{b}^{2}}c}\times \frac{{{c}^{4}}}{3{{a}^{3}}}\div \frac{4b{{c}^{3}}}{9{{a}^{2}}}
\left( v \right)\frac{1}{{{x}^{2}}-3x+2}+\frac{1}{{{x}^{2}}-5x+6}+\frac{1}{{{x}^{2}}-4x+3}          \left( vi \right)\frac{1}{x-1}+\frac{1}{x+1}+\frac{2x}{{{x}^{2}}+1}+\frac{4{{x}^{3}}}{{{x}^{4}}+1}
\left( vii \right)\frac{{{b}^{2}}-5b}{3b-4a}\times \frac{9{{b}^{2}}-16{{a}^{2}}}{{{b}^{2}}-25}\div \frac{3{{b}^{2}}+4ab}{ab+5a}          \left( viii \right)\frac{b+c}{\left( a-b \right)\left( a-c \right)}+\frac{c+a}{\left( b-a \right)\left( b-c \right)}+\frac{a+b}{\left( c-a \right)\left( c-b \right)}
\left( ix \right)\frac{b+c-a}{\left( a-b \right)\left( a-c \right)}+\frac{c+a-b}{\left( b-c \right)\left( b-a \right)}+\frac{a+b-c}{\left( c-a \right)\left( c-b \right)}          \left( x \right)\frac{\frac{{{a}^{2}}}{x-a}+\frac{{{b}^{2}}}{x-b}+\frac{{{c}^{2}}}{x-c}+a+b+c}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}}
\left( xi \right)\left( \frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}-{{b}^{2}}}-\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}} \right)\div \left( \frac{a+b}{a-b}-\frac{a-b}{a+b} \right)\times \left( \frac{a}{b}+\frac{b}{a} \right)
\left( xii \right)\frac{b+c}{bc}\left( b+c-a \right)+\frac{c+a}{ca}\left( c+a-b \right)+\frac{a+b}{ab}\left( a+b-c \right)
\left( xiii \right)\frac{{{y}^{2}}+yz+{{z}^{2}}}{\left( x-y \right)\left( x-z \right)}+\frac{{{z}^{2}}+zx+{{x}^{2}}}{\left( y-z \right)\left( y-x \right)}+\frac{{{x}^{2}}+xy+{{y}^{2}}}{\left( z-x \right)\left( z-y \right)}

উত্তর-

\[\left( i \right)\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\]

\[=\frac{c+a+b}{abc}\]

\[=\frac{a+b+c}{abc}\]

\[\left( ii \right)\frac{a-b-c}{a}+\frac{a+b+c}{a}\]

\[=\frac{a-b-c+a+b+c}{a}\]

\[=\frac{2a}{a}=2\]

\[\left( iii \right)\frac{{{x}^{2}}+{{a}^{2}}}{ab}+\frac{x-a}{ax}-\frac{{{x}^{3}}}{b}\]

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

\[=\frac{{{x}^{3}}+{{a}^{2}}x+bx-ab-a{{x}^{4}}}{abx}\]

\[\left( iv \right)\frac{2{{a}^{2}}b}{3{{b}^{2}}c}\times \frac{{{c}^{4}}}{3{{a}^{3}}}\div \frac{4b{{c}^{3}}}{9{{a}^{2}}}\]

\[=\frac{2{{a}^{2}}b}{3{{b}^{2}}c}\times \frac{{{c}^{4}}}{3{{a}^{3}}}\times \frac{9{{a}^{2}}}{4b{{c}^{3}}}\]

\[=\frac{a}{2{{b}^{2}}}\]

\[\left( v \right)\frac{1}{{{x}^{2}}-3x+2}+\frac{1}{{{x}^{2}}-5x+6}+\frac{1}{{{x}^{2}}-4x+3}\]

\[=\frac{1}{{{x}^{2}}-\left( 2+1 \right)x+2}+\frac{1}{{{x}^{2}}-\left( 3+2 \right)x+6}+\frac{1}{{{x}^{2}}-\left( 3+1 \right)x+3}\]

\[=\frac{1}{{{x}^{2}}-2x-x+2}+\frac{1}{{{x}^{2}}-3x-2x+6}+\frac{1}{{{x}^{2}}-3x-x+3}\]

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

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

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

\[=\frac{3x-6}{\left( x-1 \right)\left( x-2 \right)\left( x-3 \right)}\]

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

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

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

\[\left( vi \right)\frac{1}{x-1}+\frac{1}{x+1}+\frac{2x}{{{x}^{2}}+1}+\frac{4{{x}^{3}}}{{{x}^{4}}+1}\]

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

\[=\frac{2x}{{{x}^{2}}-1}+\frac{2x}{{{x}^{2}}+1}+\frac{4{{x}^{3}}}{{{x}^{4}}+1}\]

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

\[=\frac{2{{x}^{3}}+2x+2{{x}^{3}}-2x}{{{x}^{4}}-1}+\frac{4{{x}^{3}}}{{{x}^{4}}+1}\]

\[=\frac{4{{x}^{3}}}{{{x}^{4}}-1}+\frac{4{{x}^{3}}}{{{x}^{4}}+1}\]

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

\[=\frac{4{{x}^{7}}+4{{x}^{3}}4{{x}^{7}}-4{{x}^{3}}}{{{x}^{8}}-1}\]

\[=\frac{8{{x}^{7}}}{{{x}^{8}}-1}\]

\[\left( vii \right)\frac{{{b}^{2}}-5b}{3b-4a}\times \frac{9{{b}^{2}}-16{{a}^{2}}}{{{b}^{2}}-25}\div \frac{3{{b}^{2}}+4ab}{ab+5a}\]

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

\[=\frac{b\left( b-5 \right)}{3b-4a}\times \frac{\left( 3b+4a \right)\left( 3b-4a \right)}{\left( b+5 \right)\left( b-5 \right)}\times \frac{a\left( b+5 \right)}{b\left( 3b+4b \right)}\]

\[=a\]

\[\left( viii \right)\frac{b+c}{\left( a-b \right)\left( a-c \right)}+\frac{c+a}{\left( b-a \right)\left( b-c \right)}+\frac{a+b}{\left( c-a \right)\left( c-b \right)}\]

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

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

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

\[=\frac{0}{\left( a-b \right)\left( a-c \right)\left( b-c \right)}\]

\[=0\]

\[\left( ix \right)\frac{b+c-a}{\left( a-b \right)\left( a-c \right)}+\frac{c+a-b}{\left( b-c \right)\left( b-a \right)}+\frac{a+b-c}{\left( c-a \right)\left( c-b \right)}\]

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

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

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

\[=\frac{0}{\left( a-b \right)\left( a-c \right)\left( b-c \right)}\]

\[=0\]

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

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

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

\[=\frac{\frac{{{a}^{2}}+ax-{{a}^{2}}}{x-a}+\frac{{{b}^{2}}+bx-{{b}^{2}}}{x-b}+\frac{{{c}^{2}}+cx-{{c}^{2}}}{x-c}}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}}\]

\[=\frac{\frac{ax}{x-a}+\frac{bx}{x-b}+\frac{cx}{x-c}}{\frac{a}{x-a}+\frac{b}{x-b}+\frac{c}{x-c}}\]

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

\[=x\]

\[\left( xi \right)\left( \frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}-{{b}^{2}}}-\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}} \right)\div \left( \frac{a+b}{a-b}-\frac{a-b}{a+b} \right)\times \left( \frac{a}{b}+\frac{b}{a} \right)\]

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

\[=\frac{4{{a}^{2}}{{b}^{2}}}{\left( {{a}^{2}}-{{b}^{2}} \right)\left( {{a}^{2}}+{{b}^{2}} \right)}\div \frac{4ab}{{{a}^{2}}-{{b}^{2}}}\times \frac{{{a}^{2}}+{{b}^{2}}}{ab}\left[ \because {{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=4ab \right]\]

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

\[=1\]

\[\left( xii \right)\frac{b+c}{bc}\left( b+c-a \right)+\frac{c+a}{ca}\left( c+a-b \right)+\frac{a+b}{ab}\left( a+b-c \right)\]

\[=\frac{\left( b+c \right)\left( b+c-a \right)}{bc}+\frac{\left( c+a \right)\left( c+a-b \right)}{ca}+\frac{\left( a+b \right)\left( a+b-c \right)}{ab}\]

\[=\frac{{{b}^{2}}+bc-ab+bc+{{c}^{2}}-ac}{bc}+\frac{{{c}^{2}}+ac-bc+ac+{{a}^{2}}-ab}{ca}+\frac{{{a}^{2}}+ab-ac+ab+{{b}^{2}}-bc}{ab}\]

\[=\frac{a\left( {{b}^{2}}+2bc-ab+{{c}^{2}}-ac \right)+b\left( {{c}^{2}}+2ac-bc+{{a}^{2}}-ab \right)+c\left( {{a}^{2}}+2ab-ac+{{b}^{2}}-bc \right)}{abc}\]

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

\[=\frac{6abc}{abc}\]

\[=6\]

\[\left( xiii \right)\frac{{{y}^{2}}+yz+{{z}^{2}}}{\left( x-y \right)\left( x-z \right)}+\frac{{{z}^{2}}+zx+{{x}^{2}}}{\left( y-z \right)\left( y-x \right)}+\frac{{{x}^{2}}+xy+{{y}^{2}}}{\left( z-x \right)\left( z-y \right)}\]

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

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

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

\[=\frac{0}{\left( x-y \right)\left( y-z \right)\left( x-z \right)}\]

\[=0\]

;