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Chapter 3: Problem 86

Determine if the statement is true or false. Given \(f(x)=2 i x^{4}-(3+6 i) x^{3}+5 x^{2}+7,\) if \(a+b i\) is a zero of \(f(x)\), then \(a-b i\) must also be a zero.

Short Answer

Expert verified
False. The roots of polynomials with complex coefficients do not necessarily form conjugate pairs.

Step by step solution

01

- Identify the nature of the function

Observe the given polynomial function: \[ f(x)=2ix^{4}-(3+6i)x^{3}+5x^{2}+7 \]We need to identify whether the coefficients are real or complex. Clearly, the coefficients are complex since they include imaginary units (i).
02

- State the property of complex roots

One of the properties of polynomials with real coefficients is that if a complex number \(a + bi\) (where \(a\) and \(b\) are real numbers) is a root, its complex conjugate \(a - bi\) must also be a root. We need to see if this property holds for polynomials with complex coefficients.
03

- Examine the given function's coefficients

For the function \(f(x)\): \[ f(x)=2ix^{4}-(3+6i)x^{3}+5x^{2}+7 \]The coefficients do not satisfy the condition of having real values. Therefore, the property that pairs conjugate roots is not necessarily valid for polynomials with purely complex coefficients.
04

- Conclude the validity of the statement

Since the polynomial \(f(x)\) has complex coefficients and the roots of such polynomials do not generally follow the conjugate root theorem, the given statement is false. Thus, if \(a + bi\) is a zero of \(f(x)\), \(a - bi\) does not have to be a zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Coefficients
Complex coefficients in polynomials mean that the numbers multiplying the variable terms include imaginary units. Imaginary units are multiples of the imaginary number \(i\), where \(i\) is defined as \(\sqrt{-1}\). For instance, looking at the polynomial function \[f(x) = 2ix^4 - (3+6i)x^3 + 5x^2 + 7\], we see coefficients like \(2i\) and \((3 + 6i)\), which involve imaginary parts.

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