possible roots? please explain

Question

P(x) = -5x4 + x3 + 2x2 – 1

 

 

Use Descartes' Rule of Signs to analyze the zeros of the following. List all possibilities.

 

Answer 1

2 changes in sign, maximum 2 positive real solutions

change x to -x and count the sign changes for negative real solutions' maximum

maximum 2 negative real solutions

look at the coefficients of each term in descending powers of x

-5, +1, + 2 -1 have 2 changes -5 to +1 and +2 to -1

change x to -x gives

-5 +1 -2 -1 have 2 change, -5 to +1 and +1 to -2

(solutions, roots, zeros or x - intercepts)

imaginary solutions come in conjugate pairs, so there are either 0, 2 or 4 imaginary solutions

possibilities are

+ real - real imaginary total

0 0 4 4

0 2 2 4

2 0 2 4

2 2 0 4

they have to sum to 4 for a 4th degree polynomial