Question

complete the paragraph below. assume that a, b, and c are real numbers with a ≠ 0. the polynomial $ax^{2}+bx+c$ has roots, including any repeated roots. this follows from the.

Explanation:

Step1: Recall the degree of the polynomial

The polynomial \( ax^2 + bx + c \) (where \( a
eq0 \)) is a quadratic polynomial, and its degree is 2.

Step2: Apply the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) has exactly \( n \) roots (counting multiplicities, i.e., repeated roots) in the complex number system. For a quadratic polynomial (\( n = 2 \)), it will have 2 roots (including repeated roots). So the first blank is filled with "2" and the second blank is filled with "Fundamental Theorem of Algebra".

Answer:

The polynomial \( ax^2 + bx + c \) has \(\boldsymbol{2}\) roots, including any repeated roots. This follows from the \(\boldsymbol{\text{Fundamental Theorem of Algebra}}\).