Solve the given expansion.
$(1 + 2 x - 3 x^{2})^{4}$

Open in App

Solution

$(1 + 2 x - 3 x^{2})^{4} = [1 + (2 x - 3 x^{2})]^{4}$

$= (1 + y)^{4}$ , where $2 x - 3 x^{2} = y$

$=^{4} C_{0} +^{4} C_{1} y +^{4} C_{2} y^{2} +^{4} C_{3} y^{3} +^{4} C_{4} y^{4}$

$= 1 + 4 y + 6 y^{2} + 4 y^{3} + y^{4}$

$= 1 + 4 (2 x - 3 x^{2}) + 6 (2 x - 3 x^{2})^{2} + 4 (2 x - 3 x^{2})^{3} + (2 x - 3 x^{2})^{4}$

$= 1 + (8 x - 12 x^{2}) + 6 [^{2} C_{0} (2 x)^{2} -^{2} C_{1} (2 x) (3 x)^{2} +^{2} C_{2} (3 x^{2})^{2}]$

$+ 4 [^{3} C_{0} (2 x)^{3} -^{3} C_{1} (2 x)^{2} (3 x)^{2} +^{3} C_{2} (2 x) (3 x^{2})^{2} -^{3} C_{3} (3 x^{2})^{3}]$

$+ [^{4} C_{0} (2 x)^{4} -^{4} C_{1} (2 x)^{3} (3 x^{2}) +^{4} C_{2} (2 x)^{2} (3 x^{2})^{2} -^{4} C_{3} (2 x) (3 x^{2})^{3} +^{4} C_{4} (3 x^{2})^{4}]$

$= 1 + (8 x - 12 x^{2}) + 6 (4 x^{2} - 12 x^{3} + 9 x^{4}) + 4 (8 x^{3} - 36 x^{4} + 54 x^{5} - 27 x^{6})$

$+ (16 x^{4} - 96 x^{5} + 216 x^{6} - 216 x^{7} + 81 x^{8})$

$= 1 + (8 x - 12 x^{2}) + (24 x^{2} - 72 x^{3} + 54 x^{4}) + (32 x^{3} - 144 x^{4} + 216 x^{5} - 108 x^{6})$

$+ (16 x^{4} - 96 x^{5} + 216 x^{6} - 216 x^{7} + 81 x^{8})$

$= 81 x^{8} - 216 x^{7} + 108 x^{6} + 120 x^{5} - 74 x^{4} - 40 x^{3} + 12 x^{2} + 8 x + 1$

Suggest Corrections

Similar questions

(1 + 2 x + 3 x^{2} + . . . .)^{- 3 / 2}, (| x | < 1)

2 x^{2} - \frac{1}{2} x = 0

x

\sqrt{\frac{x - 2}{1 - 2 x}} > 1

Solve the given inequality for real x: 3(x – 1) ≤ 2 (x – 3)

x

x

\frac{2 x + \sqrt{4 x^{2} - 1}}{2 x - \sqrt{4 x^{2} - 1}} = 4

Join BYJU'S Learning Program