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Sagot :
Step 1
Given;
[tex]f(x)=3x^3+x^2-3x-1[/tex]To find the values of x
Step 2
Find a zero of the function
[tex]\begin{gathered} Using\text{ the rational rot theorem} \\ x=1\text{ is a zero} \\ Thus; \\ 3(1)^3+(1)^2-3(1)-1=3+1-3-1=0 \\ \end{gathered}[/tex]We will now use synthetic division to find the other zeroes
[tex]\begin{gathered} \mathrm{Coefficients\:of\:the\:numerator\:polynom}ial \\ 3\:\:1\:\:-3\:\:-1 \\ \mathrm{Write\:the\:problem\:in\:synthetic\:division\:format} \\ \begin{matrix}\texttt{\:\:\:1¦\:\:\:3\:\:\:1\:\:-3\:\:-1}\\ \texttt{\:\:\:\:¦\underline{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}}\\ \texttt{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}\end{matrix} \\ \mathrm{Carry\:down\:the\:leading\:coefficient,\:unchanged,\:to\:below\:the\:division\:symbol} \\ \begin{matrix}\texttt{\:\:\:1¦\:\:\:3\:\:\:1\:\:-3\:\:-1}\\ \texttt{\:\:\:\:¦\underline{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}}\\ \texttt{\:\:\:\:\:\:\:\:3\:\:\:\:\:\:\:\:\:\:\:\:}\end{matrix} \\ Multiply\:the\:carry-down\:value\:by\:the\:zero\:of\:the\:denominator,\: \\ and\:carry\:the\:result\:up\:into\:the\text{ next column} \\ 3(1)=3 \\ \begin{matrix}\texttt{\:\:\:1¦\:\:\:3\:\:\:1\:\:-3\:\:-1}\\ \texttt{\:\:\:\:¦\underline{\:\:\:\:\:\:\:3\:\:\:\:\:\:\:\:}}\\ \texttt{\:\:\:\:\:\:\:\:3\:\:\:\:\:\:\:\:\:\:\:\:}\end{matrix} \\ \end{gathered}[/tex]Repeating these steps we will have;
[tex]\begin{gathered} \begin{matrix}\texttt{\:\:\:1¦\:\:\:3\:\:\:1\:\:-3\:\:-1}\\ \texttt{\:\:\:\:¦\underline{\:\:\:\:\:\:\:3\:\:\:4\:\:\:1}}\\ \texttt{\:\:\:\:\:\:\:\:3\:\:\:4\:\:\:1\:\:\:0}\end{matrix} \\ The\text{ resulting polynomial will be} \\ 3x^2+4x+1 \end{gathered}[/tex]Factoring the resulting polynomial using the quadratic formula
[tex]\begin{gathered} 3x^2+4x+1=0 \\ x_{1,\:2}=\frac{-4\pm \sqrt{4^2-4\cdot \:3\cdot \:1}}{2\cdot \:3} \\ x_{1,\:2}=\frac{-4\pm \:2}{2\cdot \:3} \\ x_1=\frac{-4+2}{2\cdot \:3},\:x_2=\frac{-4-2}{2\cdot \:3} \\ x_{1,\:2}=\frac{-4\pm \:2}{2\cdot \:3} \\ x_1=\frac{-4+2}{2\cdot \:3},\:x_2=\frac{-4-2}{2\cdot \:3} \\ x=-\frac{1}{3},\:x=-1 \end{gathered}[/tex]Answer;
[tex]x=-\frac{1}{3},\:x=-1,\:x=1[/tex]
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