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08 Dec 2021, 06:15
Kudos
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
60% (02:40) correct
40%
(02:59)
wrong
based on 186
sessions
History
Date
Time
Result
Not Attempted Yet
If \( x = \sqrt{2} - 1\), then the value of \(x^4+4x^3+6x^2+4x+7\) is
(A) \((\sqrt{2} -1)^4\)
(B) \(4 + 4\sqrt{2}\)
(C) 6
(D) 10
(E) 12
Are You Up For the Challenge: 700 Level Questions
(A) \((\sqrt{2} -1)^4\)
(B) \(4 + 4\sqrt{2}\)
(C) 6
(D) 10
(E) 12
Are You Up For the Challenge: 700 Level Questions
10 Jul 2025, 20:10
Kudos
Bookmarks
The expression \(x^4+4x^3+6x^2+4x+7\) looks very familiar to the binomial expansion \(x^4+4x^3+6x^2+4x+1\) (terms are symmetric 1,4,6,4,1) = \((x+1)^4\).
The expression can be termed as \(x^4+4x^3+6x^2+4x+1 + 6 = (x+1)^4 + 6\).
Answer is \((\sqrt{2} - 1 + 1)^4 +6 = 2^2 +6 = 10\)
The expression can be termed as \(x^4+4x^3+6x^2+4x+1 + 6 = (x+1)^4 + 6\).
Answer is \((\sqrt{2} - 1 + 1)^4 +6 = 2^2 +6 = 10\)
08 Dec 2021, 23:04
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Bunuel
Two ways I can think of immediately are
(I) If the choices are so wide and I am in need of saving time, I will go for approximation.
Now \(x=\sqrt{2}-1=1.4-1=0.4\)
\(x^4+4x^3+6x^2+4x+7\)
As x is positive, the answer is surely greater than 7. Only D and E are left.
\(x^4+4x^3\) will make too less an effect on overall sum.
\(6x^2+4x+7=6*(0.4)^2+4*0.4+7= ~1+1.5+7=~9.5\)
So D is the answer.
(I) If the choices were close, I will analyse the terms.
Now, \(x=\sqrt{2}-1\) or \(x+1=\sqrt{2}\), so let us see if we can get the expression in same format.
\(x^4+4x^3+6x^2+4x+7\)
\(x^4+x^3+3x^3+6x^2+4x+7\)
\(x^3(x+1)+3x^3+3x^2+3x^2+4x+7\)
\(x^3(x+1)+3x^2(x+1)+3x^2+3x+x+7\)
\(x^3(x+1)+3x^2(x+1)+3x(x+1)+(x+1)+6\)
\((x^3+3x^2+3x+1)(x+1)+6\)
\((x(x^2+2x^2+1)+x^2+2x+1)(x+1)+6\)
\((x(x+1)^2+(x+1)^2)(x+1)+6\)
\((x+1)^2(x+1)(x+1)+6=(x+1)^4+6=(\sqrt{2})^4+6=4+6=10\)
D
