Question

1. what is the minimum number of real zeros a degree-10 polynomial could have?

a. 0
b. 1
c. 3
d. 2

Explanation:

Step1: Recall Polynomial Zero Rules

A polynomial of degree \( n \) has at most \( n \) real zeros (Fundamental Theorem of Algebra). For real zeros, consider the nature of even - degree polynomials. An even - degree polynomial (like degree 10, which is even) has the end - behavior that as \( x
ightarrow\pm\infty \), \( y
ightarrow+\infty \) (if the leading coefficient is positive) or \( y
ightarrow-\infty \) (if the leading coefficient is negative). But we can construct an even - degree polynomial that never crosses the \( x \) - axis. For example, consider the polynomial \( f(x)=x^{10}+1 \). The minimum value of \( x^{10} \) is 0 (when \( x = 0 \)), so \( f(0)=0 + 1=1>0 \), and for all real \( x \), \( x^{10}\geq0 \), so \( f(x)=x^{10}+1\geq1>0 \). This polynomial has no real zeros.

Step2: Evaluate the Options

We know that a degree - 10 polynomial (even degree) can have 0 real zeros. It cannot have a minimum number of real zeros less than 0, and we have shown that 0 is possible. So the minimum number of real zeros a degree - 10 polynomial could have is 0.

Answer:

a. 0