What are the possible numbers of positive, negative, and complex zeros of f(x) = −3x4 − 5x3 − x2 − 8x + 4? Select one:a. Positive: 2 or 0; negative: 2 or 0; complex: 4 or 2 or 0b. Positive: 1; negative: 3 or 1; complex: 2 or 0c. Positive: 3 or 1; negative: 1; complex: 2 or 0d. Positive: 4 or 2 or 0; negative: 2 or 0; complex: 4 or 2 or 0

Answer:b.Step-by-step explanation:We have to look at sign changes in f(x) to determine the possible positive real roots.[tex]f(x)=-3x^4-5x^3-x^2-8x+4[/tex]There is only one sign change here, between the -8x and the +4.  So that means there is only 1 possible real positive root.Now we have to look at sign changes in f(-x) to determine the possible negative real roots.[tex]f(-x)=-3x^4+5x^3-x^2+8x+4[/tex]There are 3 sign changes here.  That means there are either 3 negative roots or 3-2 = 1 negative root.  So we have:1 positive3 or 1 negativeWe need to pair them up now with all the possible combinations. If we have 1 positive and 1 negative, we have to have 2 imaginaryIf we have 1 positive and 3 negative, we have to have 0 imaginaryKeep in mind that the total number or roots--positive, negative, imaginary--have to add up to equal the degree of the polynomial.  This is a 4th degree polynomial, so we will have 4 roots.