Question

Show that the polynomial has a zero between – 2 and - 1. X4 - 6x3 - 30x2 + 48x + 82

Answer 1

Answer:

So, the polynomial has a zero between – 2 and - 1.

Step-by-step explanation:

To find the zeros (or roots) of the polynomial functions, we need to use the remainder theorem and intermediate value theorem.

So here also we will find the value of the polynomial at x=-1 and at x=-2.

So we get:

$f(-2)=(-2)^4-6(-2)^3-30(-2)^2+48(-2)+82$

$=16-\left(-48\right)-120-96+82\\=-70$

$f(-1)=(-1)^4-6(-1)^3-30(-1)^2+48(-1)+82\\=1-\left(-6\right)-30-48+82\\=11$

So here we see that the value of the polynomial at two different values of x, is opposite in sign. So this implies that there must be a zero between – 2 and - 1.