The relationship between the degree of a polynomial and the number of zeros and factors it has can be analyzed using the Fundamental Theorem of Algebra and the concept of multiplicity.
According to the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n complex zeros, taking into account multiplicity. This means that a polynomial of degree n can have n distinct zeros or fewer if some zeros have a multiplicity greater than 1.
The multiplicity of a zero of a polynomial refers to the number of times that zero appears as a root of the polynomial. For example, if a zero has a multiplicity of 2, it will contribute to the polynomial's factorization twice.
Considering the relationship between zeros and factors, every zero of a polynomial corresponds to a linear factor of the polynomial. For example, if a polynomial has a zero of (x - a), then it has a linear factor (x - a). Moreover, for each multiplicity greater than 1, it increases the power of the factor, indicating a repeated root.
Hence, in summary, the degree of a polynomial determines the maximum number of zeros that polynomial can have, and the multiplicity of these zeros influences the number of corresponding factors.
describe the relationship between the degree of a polynomial and the number of zeros and factors it has
1 answer