Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; -5 and 4+3i are zeros; f(2) = 91

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Identify the given zeros of the polynomial. Since the polynomial has real coefficients and one zero is complex (4 + 3i), its complex conjugate (4 - 3i) must also be a zero. So, the zeros are -5, 4 + 3i, and 4 - 3i.

Write the factors corresponding to each zero. For zero -5, the factor is \((x + 5)\). For zeros \(4 + 3i\) and \(4 - 3i\), the factors are \((x - (4 + 3i))\) and \((x - (4 - 3i))\) respectively.

Multiply the complex conjugate factors to get a quadratic factor with real coefficients: \[(x - (4 + 3i))(x - (4 - 3i)) = (x - 4 - 3i)(x - 4 + 3i)\] Use the difference of squares formula to simplify this product.

Express the polynomial function as \(f(x) = a(x + 5)(x^2 - 8x + 25)\), where \(a\) is a real number constant to be determined.

Use the given function value \(f(2) = 91\) to find \(a\). Substitute \(x = 2\) into the polynomial and set the expression equal to 91, then solve for \(a\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Polynomial Zeros and Complex Conjugates

For polynomials with real coefficients, non-real zeros always come in conjugate pairs. Given a zero like 4 + 3i, its conjugate 4 - 3i must also be a zero. This ensures the polynomial remains with real coefficients when expanded.

Constructing a Polynomial from Zeros

A polynomial can be formed by multiplying factors corresponding to its zeros. For zeros r1, r2, and r3, the polynomial is f(x) = a(x - r1)(x - r2)(x - r3), where 'a' is a leading coefficient determined by additional conditions.

Using Function Values to Determine Leading Coefficient

Given a specific function value like f(2) = 91, substitute x = 2 into the polynomial expression and solve for the leading coefficient 'a'. This step ensures the polynomial satisfies all given conditions.