37-46= Finding a Polynomial with Specified Zeros Find a...

$37-46=$ Finding a Polynomial with Specified Zeros Find a polynomial with integer coefficients that satisfies the given conditions. $$ \begin{array}{l}{R \text { has degree } 4 \text { and zeros } 1-2 i \text { and } 1, \text { with } 1 \text { a zero of }} \\ {\text { multiplicity } 2 .}\end{array} $$

$37-46=$ Finding a Polynomial with Specified Zeros Find a
polynomial with integer coefficients that satisfies the given
conditions.

$$
\begin{array}{l}{R \text { has degree } 4 \text { and zeros } 1-2 i \text { and } 1, \text { with } 1 \text { a zero of }} \\ {\text { multiplicity } 2 .}\end{array}
$$

Key Concepts

Integer Coefficients

When a polynomial is required to have integer coefficients, it imposes a condition on the zeros and the construction of its factors. This condition often forces the inclusion of complex conjugate pairs and requires certain choices in scaling the polynomial to ensure that, after all multiplications and expansions, every coefficient remains an integer.

Polynomial

A polynomial is an algebraic expression constituted by variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. Understanding polynomials is essential as they are used to represent a wide range of mathematical phenomena and to solve equations by finding their zeros.

Zero (Root) of a Polynomial

A zero or root of a polynomial is a solution to the equation formed when the polynomial is set equal to zero. Recognizing that a zero corresponds to a factor of the polynomial helps in constructing the polynomial by expressing it as a product of its factors.

Complex Conjugate Pairs

When a polynomial with real (or integer) coefficients has a complex zero, its complex conjugate must also be a zero. This property ensures that any terms involving complex numbers will cancel out, resulting in all coefficients being real, which is critical when constructing polynomials with integer coefficients.

Multiplicity of Zeros

The multiplicity of a zero indicates how many times a particular zero appears as a solution of the polynomial. It directly influences the factorization of the polynomial, as a zero with multiplicity greater than one contributes a repeated factor, affecting the shape and properties of the graph of the polynomial.

Constructing a Polynomial from Zeros

One common method to construct a polynomial with given zeros is to express the polynomial as a product of factors, where each factor is of the form (x - a) for a zero a. When dealing with multiplicities, the corresponding factor is raised to the appropriate exponent, and when the zeros are complex, the corresponding conjugate pairs are both included to ensure real coefficients.

Transcript

Please give Ace some feedback