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FIND THE ZEROES OF 5X^3-8X^2-5X+2

Guest May 14, 2015

Best Answer 

 #1
avatar+90084 
+10

5x^3 - 8x^2 - 5x  + 2 = 0 ..this doesn't appear to factor....let's use the "Rational Zeroes" Theorem to  find a root - if possible...

The possible roots  are ± 1, ± 2 and ± 2/5

I see that 2 is a root......let's use synthetic division to find the next polynomial

 

2       5   -8     -5     2

              10      4    -2

         5     2     -1    0

 

And the next polynomial is

5x^2 + 2x -1   = 0

This will not factor...using the on-site solver [and the Quadratic Formula] ...the other two roots are :

 

$${\mathtt{5}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{6}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{5}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{6}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{5}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.689\: \!897\: \!948\: \!556\: \!635\: \!6}}\\
{\mathtt{x}} = {\mathtt{0.289\: \!897\: \!948\: \!556\: \!635\: \!6}}\\
\end{array} \right\}$$

Here's a graph......https://www.desmos.com/calculator/hpqotand0v

 

  

CPhill  May 14, 2015
 #1
avatar+90084 
+10
Best Answer

5x^3 - 8x^2 - 5x  + 2 = 0 ..this doesn't appear to factor....let's use the "Rational Zeroes" Theorem to  find a root - if possible...

The possible roots  are ± 1, ± 2 and ± 2/5

I see that 2 is a root......let's use synthetic division to find the next polynomial

 

2       5   -8     -5     2

              10      4    -2

         5     2     -1    0

 

And the next polynomial is

5x^2 + 2x -1   = 0

This will not factor...using the on-site solver [and the Quadratic Formula] ...the other two roots are :

 

$${\mathtt{5}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{6}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{5}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{6}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{5}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.689\: \!897\: \!948\: \!556\: \!635\: \!6}}\\
{\mathtt{x}} = {\mathtt{0.289\: \!897\: \!948\: \!556\: \!635\: \!6}}\\
\end{array} \right\}$$

Here's a graph......https://www.desmos.com/calculator/hpqotand0v

 

  

CPhill  May 14, 2015

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