Give the maximum number of zeros the polynomial function may have. Use Descarte's Rule of Signs to determine how
many positive and how many negative zeros it may have.
-$$f ( x ) = x ^ { 5 } - 2.9 x ^ { 4 } - 13.51 x ^ { 3 } + 28.816 x ^ { 2 } + 37.76 x - 5.567$$
A) 5 ; 2 or 0 positive zeros; 2 or 0 negative zeros
B) $5 ; 2$ or 0 positive zeros; 3 or 1 negative zeros
C) 5 ; 3 or 1 positive zeros; 3 or 1 negative zeros
D) 5 ; 3 or 1 positive zeros; 2 or 0 negative zeros

Question

Quizplus · Accepted Answer

The polynomial has a degree of 5, so it can have a maximum of 5 zeros. Using Descartes' Rule of Signs: For positive zeros, the sign changes 3 times in $f(x)$, indicating 3 or 1 positive zeros. For negative zeros, substitute $x$ with $-x$ to get $f(-x) = -x^5 - 2.9x^4 + 13.51x^3 + 28.816x^2 - 37.76x - 5.567$, which has 2 sign changes, indicating 2 or 0 negative zeros.

Give the Maximum Number of Zeros the Polynomial Function May f(x)=x5−2.9x4−13.51x3+28.816x2+37.76x−5.567f ( x ) = x ^ { 5 } - 2.9 x ^ { 4 } - 13.51 x ^ { 3 } + 28.816 x ^ { 2 } + 37.76 x - 5.567f(x)=x5−2.9x4−13.51x3+28.816x2+37.76x−5.567

Give the Maximum Number of Zeros the Polynomial Function May $f ( x ) = x ^ { 5 } - 2.9 x ^ { 4 } - 13.51 x ^ { 3 } + 28.816 x ^ { 2 } + 37.76 x - 5.567$