Question

1. how many roots does the polynomial function $f(x)=7x^5 - 4x^4 + 5x^3 + 2x^2$ have?

○ 5 roots
○ 3 roots
○ 4 roots
○ 2 roots

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 number system.

Step2: Determine the degree of the polynomial

For the polynomial \( f(x) = 7x^5 - 4x^4 + 5x^3 + 2x^2 \), the highest power of \( x \) (the degree) is 5.

Step3: Apply the theorem to find the number of roots

By the Fundamental Theorem of Algebra, since the degree of the polynomial is 5, the polynomial has 5 roots (counting multiplicities). We can also factor the polynomial as \( f(x)=x^2(7x^3 - 4x^2 + 5x + 2) \). The root \( x = 0 \) has multiplicity 2, and the cubic factor \( 7x^3 - 4x^2 + 5x + 2 \) will have 3 roots (by the Fundamental Theorem of Algebra for degree 3 polynomials), so in total \( 2 + 3=5 \) roots (counting multiplicities). So the polynomial has 5 roots.

Answer:

A. 5 roots