Question 5 Expand (x + 1/x)^6
Class X1 - Maths -Binomial Theorem Page 167
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(x + 1/x)^6
use the formula,
(x + y)^n = nC0x^n + nC1.x^(n-1)y + nC2.x^(n-2)y² +........+ nCn.y^n
(x + 1/x)^6 = 6C0.x^6 + 6C1.x^5.1/x + 6C2x^4.1/x² + 6C3.x^3.(1/x)^3 + 6C4.x^2.(1/x)^4 + 6C5.x(1/x)^5 + 6C6.(1/x)^6
= 6C0.x^6 + 6C1.x^4 + 6C2.x^4/x^2 + 6C3.x^3 × 1/x^3 + 6C4.x^2 × 1/x^2 + 6C5.x × 1/x^5 + 6C6.1/x^6
= (6!/6!)x^6 + (6!/5!)x^4 + {6×5/2}x^2 + {6×5×4/6}+{6×5/2}/x^2 + 6×1/x^4 + (6!/6!)/x^6
= x^6 + 6x^4 + 15x^2 + 20 + 15/x^2 + 6/x^4 + 1/x^6
use the formula,
(x + y)^n = nC0x^n + nC1.x^(n-1)y + nC2.x^(n-2)y² +........+ nCn.y^n
(x + 1/x)^6 = 6C0.x^6 + 6C1.x^5.1/x + 6C2x^4.1/x² + 6C3.x^3.(1/x)^3 + 6C4.x^2.(1/x)^4 + 6C5.x(1/x)^5 + 6C6.(1/x)^6
= 6C0.x^6 + 6C1.x^4 + 6C2.x^4/x^2 + 6C3.x^3 × 1/x^3 + 6C4.x^2 × 1/x^2 + 6C5.x × 1/x^5 + 6C6.1/x^6
= (6!/6!)x^6 + (6!/5!)x^4 + {6×5/2}x^2 + {6×5×4/6}+{6×5/2}/x^2 + 6×1/x^4 + (6!/6!)/x^6
= x^6 + 6x^4 + 15x^2 + 20 + 15/x^2 + 6/x^4 + 1/x^6
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