Question

Tell the maximum number of zeros that's a 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.

f(x)=9x^8-8x^6-8x+3​

Answer 1

A polynomial function can have a maximum number of zeros equal to its degree. For the given polynomial f(x) = 9x^8 - 8x^6 - 8x + 3, using Descartes' Rule of Signs, it may have up to 2 positive and 2 negative real zeros.

Step-by-step explanation:

The maximum number of zeros a polynomial function may have is equal to its degree. For the polynomial function f(x) = 9x^8 - 8x^6 - 8x + 3, the degree is 8, which means it can have at most 8 real zeros. To use Descartes' Rule of Signs, we count the number of sign changes in the polynomial.

For positive real zeros, we look at the function as it is given:

• 
  
• 9x^8 to -8x^6: one sign change
• 
  
• -8x^6 to -8x: no sign change
• 
  
• -8x to +3: one sign change

Thus, there can be a maximum of 2 positive real zeros for this polynomial.

To determine the number of negative real zeros, we look at the polynomial with x replaced by -x:

• 
  
• 9(-x)^8 to -8(-x)^6 has no sign change since both terms are positive.
• 
  
• -8(-x)^6 to -8(-x) has one sign change since it goes from a positive to negative.
• 
  
• -8(-x) to +3 has one sign change.

This yields a maximum of 2 negative real zeros.

Therefore, this polynomial may have up to 2 positive real zeros and 2 negative real zeros.