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

This question relates to the Fundamental Theorem of Algebra.
The theorem states that any non-constant single-variable polynomial with complex coefficients has at least one complex root.
A direct consequence of this theorem is that a polynomial of degree $n$ has exactly $n$ roots (or zeros) in the complex number system, counting multiplicities.
The degree of a polynomial is the highest exponent of its variable.
In the given polynomial, $p(x) = 4x^{11} - 8x^9 + 7x^5 + 6x^3 + 4x^2 + 5x + 6$.
The highest power of $x$ is 11.
Therefore, the degree of the polynomial $p(x)$ is 11.
According to the Fundamental Theorem of Algebra, the number of zeros of this polynomial is equal to its degree.
Thus, the number of zeros is 11.