Determine the number of possible positive and negative real zeros for the given function.
-$$f ( x ) = - 8 x ^ { 7 } - 7 x ^ { 4 } - 6 x ^ { 3 } + 4 x ^ { 2 } + 6 x + 5$$
A) Positive: 1; Negative: 4 or 2
B) Positive: 4 or 2; Negative: 1
C) Positive: 4; Negative: 1
D) Positive: 1; Negative: 4

Question

Quizplus · Accepted Answer

The number of possible positive real zeros of a polynomial function is determined by the number of sign changes in the coefficients of the terms when arranged in descending order of their degrees. For the given function, there is only 1 sign change (from -6x^3 to +4x^2), indicating there could be 1 positive real zero. The number of possible negative real zeros is determined by applying Descartes' Rule of Signs to the function with x replaced by -x. After substitution, the function becomes $8x^7 - 7x^4 + 6x^3 + 4x^2 - 6x + 5$, which has 4 sign changes, indicating there could be 4 or 2 negative real zeros (subtracting 2 for each subsequent possibility).

Determine the Number of Possible Positive and Negative Real Zeros f(x)=−8x7−7x4−6x3+4x2+6x+5f ( x ) = - 8 x ^ { 7 } - 7 x ^ { 4 } - 6 x ^ { 3 } + 4 x ^ { 2 } + 6 x + 5f(x)=−8x7−7x4−6x3+4x2+6x+5

Determine the Number of Possible Positive and Negative Real Zeros $f ( x ) = - 8 x ^ { 7 } - 7 x ^ { 4 } - 6 x ^ { 3 } + 4 x ^ { 2 } + 6 x + 5$