Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational
roots.
-The concentration, in parts per million, of a particular drug in a patient's blood x hours after the drug is administered is given by the function
$f ( x ) = - x ^ { 4 } + 10 x ^ { 3 } - 36 x ^ { 2 } + 48 x$
How many hours after the drug is administered will it be eliminated from the bloodstream.
A) 4 hours
B) 12 hours
C) 5 hours
D) 16 hours

Question

Quizplus · Accepted Answer

The drug will be eliminated from the bloodstream when the concentration is 0, i.e., when $f(x) = 0$. The given polynomial is $f(x) = -x^4 + 10x^3 - 36x^2 + 48x$. Factoring out $x$, we get $x(-x^3 + 10x^2 - 36x + 48) = 0$. The first root is $x = 0$, but this does not represent a time after administration. To find the non-zero roots, we can use synthetic division to test the possible rational roots, which are factors of 48. Testing $x = 1, 2, 3, 4$, etc., we find that $x = 4$ is a root since $4^3 \cdot (-1) + 10 \cdot 4^2 - 36 \cdot 4 + 48 = 0$. Therefore, the drug will be eliminated from the bloodstream 4 hours after it is administered.

Solve the Polynomial Equation f(x)=−x4+10x3−36x2+48xf ( x ) = - x ^ { 4 } + 10 x ^ { 3 } - 36 x ^ { 2 } + 48 xf(x)=−x4+10x3−36x2+48x

Solve the Polynomial Equation $f ( x ) = - x ^ { 4 } + 10 x ^ { 3 } - 36 x ^ { 2 } + 48 x$