$f(x) = 2x^3 - 5x^2 + 7x + 10$
Given that $2x - 3$ is a factor, solve $f(x) = 0$ completely.
I have tried using a division of polynomials method to give a quadratic, but this gave a remainder of $16$. I have also used online calculators which suggest that there is no simple solution. Is there an error or should I be using a different method?
Thanks in advance.
On
Like many users in the comments I have entered the question into Wolfram Alpha and it did indeed give two complex roots and one real root, albeit not the $\frac{3}{2}$ that would be suggested by using factor theorem if the source question had been correct. It does look like the source question was flawed. Thanks for all the support.
Link to Wolfram Alpha solution page
If we put $x=\frac 32$ into $f(x)$ then we can see that \begin{align}f\left(\frac 32\right)&=2\left(\frac 32\right)^3−5\left(\frac 32\right)^2+7\left(\frac 32\right)+10\\ &=\frac {27}4-\frac {45}4+\frac{21}2+10\\ &=16\neq 0\end{align}
As $f\left(\frac 32\right)$ is not equal to $0$, then $2x-3$ is not a root.
This appears to be a homework/exam style question, are you sure you've written down $f(x)$ correctly? Either that, or the person who wrote the question mis-wrote $f(x)$ in the first place.