The number of solutions of the equation \( \dfrac{1}{2}(x^3+1)=(2x-1)^{⅓}\) is
\(0\)
\(6\)
\(9\)
\(∞\)
\(3\)
The Correct Option is C
Solution and Explanation
\(\dfrac{1}{2}(x^3+1)=(2x-1)^{⅓}\)
\(⇒\)\([\dfrac{1}{2}(x^3+1)]^{3}=2x-1\)
\(⇒[\dfrac{1}{8}(x^9+3x^6+3x^3+1)]=2x-1\)
\(⇒[\dfrac{1}{8}(x^9+3x^6+3x^3+1)]=2x-1\)
\(⇒[\dfrac{1}{8}(x^9+3x^6+3x^3-2x+9)=0\)
Hence this polynomial of degree \(9\) has \(9\) solutions.(_Ans.)
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