Tell the maximum number of zeros that the polynomial functio

Tell the maximum number of zeros that the polynomial function may have. Then use Descartes\' Rule of Signs to determine how many positive and how many negative real zeros the polynomial function may have. Do not attempt to find the zeros f(x) = 8x^7 + 9x^6 + x^4 + x + 7 What is the maximum number of zeros that this polynomial function can have? How many positive real zeros can the function have? (Use a comma to separate answers as needed.) How many negative real zeros can the function have? (Use a comma to separate answers as needed.)

Solution

f(x) = 8x^7 + 9x^6 + x^4 + x+ 7

maximum number of zeros are determined by the highest exponent of the function

the polynomial can have maximum of 7 zeros

applying descartes rule of signs

8x^7 + 9x^6 + x^4 + x+ 7

there are no sign change going from left to right

so there are no positive real zeros

f(-x) = 8(-x)^7 + 9(-x)^6 +(-x)^4 +(-x) + 7

f(-x) = -8x^7 +9x^6 + x^4 - x + 7

going from left to right

there are total 3 sign changes

therefore, the function can have 3,1 negative zeros