According to the Fundamental Theorem of Algebra, which polynomial function has exactly 6 roots?
A. [tex]f(x)=5 x^4+10 x^2+2[/tex]
B. [tex]f(x)=5 x^5+3 x^4+12 x^3+7 x^2-2 x+15[/tex]
C. [tex]f(x)=6 x^5+x^3-4 x^2+x-5[/tex]
D. [tex]f(x)=7 x^6+3 x^3+12[/tex]
Answer (1)
The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots.
Identify the degree of each polynomial.
The polynomial f ( x ) = 5 x 4 + 10 x 2 + 2 has degree 4.
The polynomial f ( x ) = 5 x 5 + 3 x 4 + 12 x 3 + 7 x 2 − 2 x + 15 has degree 5.
The polynomial f ( x ) = 6 x 5 + x 3 − 4 x 2 + x − 5 has degree 5.
The polynomial f ( x ) = 7 x 6 + 3 x 3 + 12 has degree 6.
Therefore, the polynomial with exactly 6 roots is f ( x ) = 7 x 6 + 3 x 3 + 12 .
Explanation
Understanding the Fundamental Theorem of Algebra According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex roots, counted with multiplicity. We need to identify the polynomial function with exactly 6 roots.
Determining the Degree of Each Polynomial The degree of a polynomial is the highest power of the variable in the polynomial. We will determine the degree of each polynomial function.
Degree of the First Polynomial The degree of f ( x ) = 5 x 4 + 10 x 2 + 2 is 4.
Degree of the Second Polynomial The degree of f ( x ) = 5 x 5 + 3 x 4 + 12 x 3 + 7 x 2 − 2 x + 15 is 5.
Degree of the Third Polynomial The degree of f ( x ) = 6 x 5 + x 3 − 4 x 2 + x − 5 is 5.
Degree of the Fourth Polynomial The degree of f ( x ) = 7 x 6 + 3 x 3 + 12 is 6.
Identifying the Polynomial with 6 Roots According to the Fundamental Theorem of Algebra, the polynomial with degree 6 has exactly 6 roots. Therefore, the polynomial function with exactly 6 roots is f ( x ) = 7 x 6 + 3 x 3 + 12 .
Examples
Understanding the degree of a polynomial and the Fundamental Theorem of Algebra is crucial in many areas, such as engineering, physics, and computer science. For example, when designing a bridge, engineers use polynomial functions to model the load distribution and ensure the structure's stability. The roots of these polynomials help determine critical points where the load is maximum or minimum, ensuring the bridge can withstand the forces acting upon it. Similarly, in computer graphics, polynomial functions are used to create smooth curves and surfaces, and understanding their roots helps in rendering realistic images.
