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Section 1.1 : Review : Functions

10. Determine all the roots of \(f\left( x \right) = {x^5} - 4{x^4} - 32{x^3}\).

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Set the function equal to zero and factor the left side.

\[{x^5} - 4{x^4} - 32{x^3} = {x^3}\left( {{x^2} - 4x - 32} \right) = {x^3}\left( {x - 8} \right)\left( {x + 4} \right) = 0\]

After factoring we can see that the three roots of this function are,

\[x = - 4,\,\,\,\,\,\,x = 0,\,\,\,\,\,\,x = 8\]