Question

Show that p(x)=2x3−5x2−10x+5 has a real root.

Answer 1

Let's pick an arbitrary value of 5.

$P(5) = 2(5)^{3} - 5(5)^{2} - 10(5) + 5$
$= 2(125) - 125 - 50 + 5$
$= 250 - 125 - 50 + 5$
$= 80 > 0$

Let's pick another arbitrary value: 1

$P(1) = 2 - 5 - 10 + 5$
$= -8 < 0$

Between these two values, the polynomial is continuous. This means for the two polynomial points to be satisfied, there must be at least one root between these two points.

We can either use the Intermediate Value Theorem or Newton's method to make a better approximation from there.