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.

$$
P \text { has degree } 2 \text { and zeros } 1+i \sqrt{2} \text { and } 1-i \sqrt{2}
$$

Key Concepts

Complex Conjugate Root Theorem

This theorem states that if a polynomial has real coefficients and a complex number is a zero, then its complex conjugate is also a zero. This concept is crucial when constructing polynomials with complex roots because ensuring the coefficients are real (or integer) requires including both a complex number and its conjugate as roots.

Factor Theorem

The Factor Theorem is the concept that if a number r is a zero of a polynomial, then (x - r) is a factor of that polynomial. This allows one to build the polynomial by taking the product of factors corresponding to each of the given zeros.

Polynomial Construction from Roots

This concept involves constructing a polynomial by multiplying linear factors derived from each root. For a specified degree, such as a quadratic, the polynomial can be written as the product of two factors corresponding to the two roots. Expanding these factors gives the polynomial in standard form with coefficients that reflect the provided roots.

Integer Coefficient Polynomials

When a polynomial is required to have integer coefficients, any irrational or complex roots must appear in conjugate pairs. This ensures that when the factors are expanded, the irrational or imaginary parts cancel out, resulting in integer (or at least rational) coefficients. This requirement influences the selection and pairing of roots when constructing the polynomial.

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