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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 29
Capitolo 4, Problema 29

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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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.
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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.
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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.
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