$h(x)=2 x^2+26$
The fundamental theorem of algebra tells you that this polynomial function has $\square$ roots.
Answer (1)
The polynomial function is h ( x ) = 2 x 2 + 26 .
The degree of the polynomial is 2.
By the fundamental theorem of algebra, the polynomial has 2 roots.
The polynomial function h ( x ) has 2 roots.
Explanation
Understanding the Problem The given polynomial function is h ( x ) = 2 x 2 + 26 . The fundamental theorem of algebra states that a polynomial of degree n has n complex roots, counted with multiplicity.
Determining the Degree of the Polynomial The degree of the polynomial h ( x ) = 2 x 2 + 26 is 2, since the highest power of x is 2.
Applying the Fundamental Theorem of Algebra According to the fundamental theorem of algebra, a polynomial of degree n has n roots. Therefore, the polynomial h ( x ) has 2 roots.
Conclusion The fundamental theorem of algebra tells us that the polynomial function h ( x ) = 2 x 2 + 26 has 2 roots.
Examples
Understanding the number of roots of a polynomial is crucial in many areas, such as physics and engineering, where polynomials are used to model various phenomena. For example, in electrical engineering, the roots of a characteristic polynomial determine the stability of a circuit. Knowing that a polynomial of degree n has n roots helps engineers design stable and reliable systems. This concept also extends to control systems, where the roots of the system's transfer function determine its behavior and stability.
