Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given
function.
-$$f ( x ) = x ^ { 5 } - 3.2 x ^ { 4 } - 19.58 x ^ { 3 } + 3 x ^ { 2 } + 49.21 x - 13.538$$
A) 3 or 1 positive zeros, 2 or 0 negative zeros
B) 2 or 0 positive zeros, 2 or 0 negative zeros
C) 3 or 1 positive zeros, 3 or 1 negative zeros
D) 2 or 0 positive zeros, 3 or 1 negative zeros

Question

Quizplus · Accepted Answer

Descartes's Rule of Signs indicates that the number of positive real zeros is determined by the number of sign changes in the polynomial's terms. For $f(x)$, the sign changes are: $+$ to $-$, $-$ to $+$, and $+$ to $-$, which gives 3 sign changes, indicating 3 or 1 positive zeros. For negative zeros, consider $f(-x)$: $-x^5 - 3.2x^4 + 19.58x^3 + 3x^2 - 49.21x - 13.538$, which has 2 sign changes, indicating 2 or 0 negative zeros.

Use Descartes's Rule of Signs to Determine the Possible Number f(x)=x5−3.2x4−19.58x3+3x2+49.21x−13.538f ( x ) = x ^ { 5 } - 3.2 x ^ { 4 } - 19.58 x ^ { 3 } + 3 x ^ { 2 } + 49.21 x - 13.538f(x)=x5−3.2x4−19.58x3+3x2+49.21x−13.538

Use Descartes's Rule of Signs to Determine the Possible Number $f ( x ) = x ^ { 5 } - 3.2 x ^ { 4 } - 19.58 x ^ { 3 } + 3 x ^ { 2 } + 49.21 x - 13.538$