In Exercises 61–62, find an nth-degree polynomial function...

In Exercises 61–62, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, graph the function and verify the real zeros and the given function value. $$ n=3 ; 2 \text { and } 2-3 i \text { are zeros; } f(1)=-10 $$

In Exercises 61–62, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, graph the function and verify the real zeros and the given function value.
$$
n=3 ; 2 \text { and } 2-3 i \text { are zeros; } f(1)=-10
$$

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Key Concepts

Determining the Leading Coefficient

Additional conditions, such as a specified function value at a particular point, are used to determine the unknown multiplicative constant (leading coefficient) in the polynomial. This involves substituting the given value into the polynomial and solving for the constant.

Degree of the Polynomial

The degree of a polynomial is determined by the number of zeros, counting multiplicity. In cases where some zeros are complex, they still contribute to the overall degree, allowing the construction of a polynomial that meets the given degree requirement.

Complex Conjugate Theorem

When constructing polynomials with real coefficients, if a complex number is a zero, its complex conjugate is also a zero. This ensures that the factors corresponding to the complex roots pair up to produce a quadratic factor with real coefficients.

Factorization of Polynomials

A polynomial can be expressed as a product of factors corresponding to its zeros. Each zero contributes a factor of the form (x - zero), and the complete polynomial is formed by multiplying these factors together, possibly scaled by a constant.