Question

If \(p(x) = 4x^{11} - 8x^9 + 7x^5 + 6x^3 + 4x^2 + 5x + 6\), then the number of zeros is ________.

• A. 7
• B. 6
• C. 9
• D. 11

Hint:

The degree of a polynomial gives the maximum number of real or complex zeros it can have.

Answer 1

The Correct Option is D

Step 1: Polynomial Degree.
The given polynomial is: \[ p(x) = 4x^{11} - 8x^9 + 7x^5 + 6x^3 + 4x^2 + 5x + 6. \] The degree of the polynomial is determined by the highest power of \(x\), which is 11. Hence, the polynomial is of degree 11.

Step 2: Maximum Number of Zeros.
According to the Fundamental Theorem of Algebra, a polynomial of degree \(n\) has at most \(n\) real or complex roots (zeros). Therefore, the polynomial can have at most 11 zeros.

Step 3: Conclusion.
Since the degree of the polynomial is 11, the number of zeros is at most 11. Therefore, the number of zeros is 11.

Final Answer: 11.