Write a polynomial f(x) that satisfies the given conditions....

Write a polynomial $f(x)$ that satisfies the given conditions. (See Example 6 )
Polynomial of lowest degree with real coefficients and with zeros $7-4 i$ (multiplicity 1 ) and 0 (multiplicity 4 ).

Key Concepts

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable that appears in the polynomial and represents the total sum of the multiplicities of all its zeros. Finding a polynomial of the lowest degree that satisfies given conditions means using the minimal number of factors corresponding to the zeros, including complex conjugate pairs and repeated zeros.

Complex Conjugate Root Theorem

This theorem states that if a polynomial has real coefficients, any non-real complex zero must occur in conjugate pairs. That is, if a + bi is a zero, then its complex conjugate a - bi must also be a zero. This is crucial in ensuring that the polynomial coefficients remain real when constructing the polynomial from its zeros.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears as a solution of the polynomial equation. A zero with multiplicity greater than one indicates that the corresponding factor appears repeatedly in the factorization. Multiplicity affects the degree of the polynomial and the behavior of the graph at that zero, such as touching or crossing the x-axis.

Polynomial Construction from Zeros

Constructing a polynomial from its zeros involves creating a product of factors corresponding to each zero raised to the power of its multiplicity. Each factor has the form (x - zero). For polynomials with real coefficients, when a complex zero is included, its conjugate must also be included, resulting in quadratic factors that yield real coefficients upon multiplication.

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