द्विपद प्रमेय की सहायता से सरल कीजिए - `(x+1)^(5)+(x-1)^(5)`

Explanation

`(x+1)^(5)=sum_(r=0)^(5).^(5)C_(r)x^(5-r)*1^(r)=sum_(r=0)^(5).^(5)C_(r)*x^(5-r)`
`rArr(x+1)^(5)=.^(5)C_(0)x^(5)+.^(5)C_(1)*x^(4)+.^(5)C_(2)x^(3)+.^(5)C_(3)x^(2)+.^(5)C_(4)x+.^(5)C_(5)` . . . (1)
और `(x-1)^(5)=sum_(r=0)^(5).^(5)C_(r)x^(5-r)(-1)^(r)`
`=.^(5)C_(0)x^(5)+.^(5)C_(1)x^(4)*(-1)^(1)+.^(5)C_(2)x^(3)(-1)^(2)+.^(5)C_(3)x^(2)*(-1)^(3)+.^(5)C_(4)x(-1)^(4)+.^(5)C_(5)(-1)^(5)`
`rArr(x-1)^(5)=.^(5)C_(0)x^(5)-.^(5)C_(1)x^(4)+.^(5)C_(2)x^(3)-.^(5)C_(3)x^(2)+.^(5)C_(4)*x-.^(5)C_(5)` . . . (2)
समीकरण (1) और (2) को जोड़ने पर
`(x+1)^(5)+(x-1)^(5)=2[.^(5)C_(0)x^(5)+.^(5)C_(2)x^(3)+.^(5)C_(4)*x]`
`=2[1*x^(5)+10*x^(3)+5*x]`
`=2(x^(5)+10x^(3)+5x)`