Use Descartes' Rule of Signs to determine the possible number of positive and negative real zeros for the polynomial
function.
-$f ( x ) = 3 x ^ { 6 } - 9 x ^ { 5 } + x ^ { 4 } - 1.5 x ^ { 3 } + 18$
A)4 or 2 or 0 positive zeros, 1 negative zeros
B)4 or 2 or 0 positive zeros, 0 negative zeros
C)4 positive zeros, 0 negative zeros
D)4 or 2 positive zeros, 0 negative zeros

Question

Quizplus · Accepted Answer

Descartes' Rule of Signs for positive zeros counts the sign changes in the original polynomial: $3x^6 - 9x^5 + x^4 - 1.5x^3 + 18$ has 3 sign changes, indicating 4 or 2 or 0 positive zeros. For negative zeros, substitute $x$ with $-x$ to get $3(-x)^6 - 9(-x)^5 + (-x)^4 - 1.5(-x)^3 + 18$, which simplifies to $3x^6 + 9x^5 + x^4 + 1.5x^3 + 18$, showing no sign changes, indicating 0 negative zeros.

Use Descartes' Rule of Signs to Determine the Possible Number f(x)=3x6−9x5+x4−1.5x3+18f ( x ) = 3 x ^ { 6 } - 9 x ^ { 5 } + x ^ { 4 } - 1.5 x ^ { 3 } + 18f(x)=3x6−9x5+x4−1.5x3+18

Use Descartes' Rule of Signs to Determine the Possible Number $f ( x ) = 3 x ^ { 6 } - 9 x ^ { 5 } + x ^ { 4 } - 1.5 x ^ { 3 } + 18$