Question

how many roots, real or complex, does the polynomial $7 + 5x^4 - 3x^2$ have in all? 5 3 4 7

Explanation:

Step1: Recall the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) has exactly \( n \) roots (counting multiplicities) in the complex plane.

Step2: Determine the degree of the polynomial

First, rewrite the polynomial \( 7 + 5x^4 - 3x^2 \) in standard form (descending powers of \( x \)): \( 5x^4 - 3x^2 + 7 \). The highest power of \( x \) (the degree) is 4.

Step3: Apply the Fundamental Theorem of Algebra

By the Fundamental Theorem of Algebra, a polynomial of degree 4 has 4 roots (real or complex, counting multiplicities).

Answer:

4