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Question
Expand `(2x^2 -3sqrt(1 - x^2))^4 + (2x^2 + 3sqrt(1 - x^2))^4`
Solution
Taking 2x2 as a and `3sqrt(1 - x^2)` as b
We have (a – b)4 + (a + b)4
Now (a – b)4 = 4C0 a3(– b) + 4C2(a2)(– b)2 + 4C3(a)(– b)3 + 4C4(– b)4
4C0 = 1 = 4C4 ; 4C1 = 4 = 4C3 ; 4C2 = `(4 xx 3)/(2 xx 1)` = 6
= a4 – 4a3b + 6a2b2 – 4ab3 + b4
Similarly (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
∴ (a – b)4 + (a + b)4 = 2[a4 + 6a2b2 + b4]
Substituting the value of a and b we get
`2[(2x^2)^4 + 6(2x^2)^2 (3sqrt(1 - x^2))^2 + (3 sqrt(1 - x^2))^4]`
= 2[16x8 + 6(4x4)(9(1 – x2)) + 81(1 – x2)2]
= 2[16x8 + 216x4(1 – x2) + 81(1 – x2)2]
= 2[16x8 + 216x4 – 216x6 + 81 + 81x4 – 162x2]
= 2[16x8 – 216x6 + 297x4 – 162x2 + 81]
= 32x8 – 432x6 + 594x4 – 324x2 + 162
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