Find š¹(š„), if š(š„ ā 2) = š„3 + 3š„2 ā 3š„ + 7
To find š¹(š„), we need to substitute (š„ + 2) in place of š„ in the equation š(š„ ā 2) = š„3 + 3š„2 ā 3š„ + 7:
š(š„ ā 2) = š„3 + 3š„2 ā 3š„ + 7
š((š„ + 2) ā 2) = (š„ + 2)3 + 3(š„ + 2)2 ā 3(š„ + 2) + 7 (substituting š„ + 2 in place of š„)
š(š„) = (š„ + 2)3 + 3(š„ + 2)2 ā 3(š„ + 2) + 7
Expanding and simplifying:
š(š„) = š„3 + 9š„2 + 23š„ + 17
Therefore, š¹(š„) = š„3 + 9š„2 + 23š„ + 17.
To find š¹(š„), we need to substitute š„ ā 2 into š(š„ ā 2) in the equation š(š„ ā 2) = š„3 + 3š„2 ā 3š„ + 7.
So, substitute š„ ā 2 for š„ in the equation:
š(š„ ā 2) = (š„ ā 2)3 + 3(š„ ā 2)2 ā 3(š„ ā 2) + 7
Now, let's simplify this expression.
First, expand (š„ ā 2)3 using the binomial theorem. This gives us:
(š„ ā 2)3 = š„3 ā 3š„2(2) + 3š„(2)2 ā 2(2)3 = š„3 ā 6š„2 + 12š„ ā 8
Next, expand 3(š„ ā 2)2. This gives us:
3(š„ ā 2)2 = 3(š„2 ā 4š„ + 4) = 3š„2 ā 12š„ + 12
Similarly, expand -3(š„ ā 2). This gives us:
-3(š„ ā 2) = -3š„ + 6
Now, substitute these expressions back into the equation:
š(š„ ā 2) = (š„ ā 2)3 + 3(š„ ā 2)2 ā 3(š„ ā 2) + 7
š(š„ ā 2) = š„3 ā 6š„2 + 12š„ ā 8 + 3š„2 ā 12š„ + 12 ā 3š„ + 6 + 7
Combine like terms:
š(š„ ā 2) = š„3 ā 6š„2 + 3š„2 + 12š„ ā 12š„ ā 3š„ ā 8 + 12 + 6 + 7
š(š„ ā 2) = š„3 + 7
Therefore, š(š„) = š¹(š„ ā 2) = š„3 + 7.