How do you find all the zeros of P(x) = 4x^3 - 5x^2 + 3x - 10?

Mar 31, 2016

The only Real root I could find is at (approximately) $x = 1.68515015$

Perhaps the complex roots could be approximated by using the quadratic formula on $4 {x}^{3} - 5 {x}^{2} + 3 x - 10 \div \left(x - 1.6515015\right)$

Explanation:

Based on the graph of this function, we can see that there is only one Real root:
graph{4x^3-5x^2+3x-10 [-18.24, 46.7, -29.57, 2.9]}
We could attempt to use the Rational Root Theorem and test
$\textcolor{w h i t e}{\text{XXX")x=-abs("factors of "10)/abs("factors of } 4}$ for $P \left(x\right) = 0$
but as demonstrated in the table below, none of these work:

As a final attempt I used the following code to apply the Newton Method which gave the approximation shown above:

'Newton.Bas
' using the Newton method to approximate root
' of 4x^3-5x^2+3x-10
' Alan P./March 2016

lo = 5/4
hi = 2

while (hi-lo) > 0.0001
mid = (lo+hi)/2
if P(mid) < 0 then
lo = mid
else
hi = mid
end if
wend

print "At x= "; mid; " P(x)= ";P(mid)

print
print
print " (Alt-F4) to end"

function P(x)
P=4x^3-5x^2+3*x-10
end function