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=4; i and 3i are zeros; f(-1) = 20

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Identify the given zeros of the polynomial. Since the polynomial has real coefficients and the zeros include complex numbers $i$ and \$3i$, their conjugates $-i$ and $-3i$ must also be zeros. So the zeros are $i$, $-i$, $3i$, and $-3i$.

Write the factors corresponding to each zero. For zeros $i$ and $-i$, the factor is $(x - i)(x + i) = x^{2} + 1$. For zeros \$3i$ and $-3i$, the factor is $(x - 3i)(x + 3i) = x^{2} + 9$.

Form the polynomial as the product of these quadratic factors multiplied by a constant $a$: $f(x) = a(x^{2} + 1)(x^{2} + 9)$.

Use the given function value $f(-1) = 20$ to find the constant $a$. Substitute $x = -1$ into the polynomial: $f(-1) = a((-1)^{2} + 1)((-1)^{2} + 9) = a(1 + 1)(1 + 9) = a(2)(10) = 20$.

Solve for $a$ from the equation $20 = 20a$ to find the value of $a$. Then write the final polynomial function $f(x)$ with this value of $a$.

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주요 개념

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

Complex Conjugate Root Theorem

For polynomials with real coefficients, non-real complex roots always come in conjugate pairs. If i is a root, then -i must also be a root. This ensures the polynomial remains with real coefficients when expanded.

Constructing Polynomials from Roots

A polynomial can be formed by multiplying factors corresponding to its roots. For roots r, the factor is (x - r). Given roots i, -i, 3i, and -3i, the polynomial is the product of (x - i)(x + i)(x - 3i)(x + 3i), which simplifies to a polynomial with real coefficients.

Using a Given Function Value to Find Leading Coefficient

After forming the polynomial with unknown leading coefficient a, substitute the given x-value (here, x = -1) and set the polynomial equal to the given function value (f(-1) = 20). Solving for a determines the exact polynomial.