Question

F(x) = 3x^3 + x^2 – 3x - 1Find the values of x

Answer 1

Step 1

Given;

`f(x)=3x^3+x^2-3x-1`

To find the values of x

Step 2

Find a zero of the function

`\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}`

We will now use synthetic division to find the other zeroes

`\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}`

Repeating these steps we will have;

`\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}`

Factoring the resulting polynomial using the quadratic formula

`\begin{gathered} 3x^2+4x+1=0 \\ x_(1,\:2)=(-4\pm √(4^2-4\cdot \:3\cdot \:1))/(2\cdot \:3) \\ x_(1,\:2)=(-4\pm \:2)/(2\cdot \:3) \\ x_1=(-4+2)/(2\cdot \:3),\:x_2=(-4-2)/(2\cdot \:3) \\ x_(1,\:2)=(-4\pm \:2)/(2\cdot \:3) \\ x_1=(-4+2)/(2\cdot \:3),\:x_2=(-4-2)/(2\cdot \:3) \\ x=-(1)/(3),\:x=-1 \end{gathered}`

Answer;

`x=-(1)/(3),\:x=-1,\:x=1`