Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Reasoning The imaginary number \(2 i\) is a zero of \(f(x)=x^{3}-2 i x^{2}-4 x+8 i\) but the complex conjugate of \(2 i\) is not a zero of \(f(x)\). Is this a contradiction of the conjugate pairs statement on page 317 ? Explain.

Step-by-Step Solution

Answer

There is no contradiction in the given situation. The rule about conjugate pairs applies when the coefficients of the polynomial are all real numbers. In the given polynomial, as there is a complex coefficient, the rule does not apply, hence it does not necessitate that the conjugate of \(2i\) is a zero of the polynomial. Therefore, the complex conjugate of \(2i\) doe not need to be a root of the given polynomial \(f(x).\)

Step 1: Identify the Zeros

Examine the given polynomial function \(f(x)=x^{3}-2 i x^{2}-4 x+8i.\) It is given that \(2 i\) is a zero of the equation. This means that by inserting \(2 i\) into \(f(x),\) result should be \(0.\)

Step 2: Analyze the Real and Imaginary Parts of the Polynomial

Look at the coefficients of the polynomial terms. There are real numbers as well as complex numbers. A crucial observation to make is that the coefficients of this polynomial are not all real numbers but there is also an imaginary number.

Step 3: Examine the Conjugate

Now we consider the conjugate \(-2i.\) Normally, as per conjugate root theorem, if the coefficients of the polynomial are all real numbers, any non-real roots must occur in complex conjugate pairs. But, the polynomial here has a complex coefficient. Hence, the theorem does not apply in this case.

Step 4: Conclusion

Based on aforementioned observations, there is no contradiction in the given scenario. \(2 i\) would have guaranteed \(-2 i\) as a root if all the coefficients of the polynomial were real numbers, but here it is not the case. Hence, complex conjugate of \(2i\) does not have to be a zero of \(f(x).\)

Key Concepts

Understanding Imaginary Numbers

To grasp the world of complex numbers, we must first understand what an imaginary number is. These numbers are crucial for solving equations that have no real number solutions. An imaginary number is defined as any real number multiplied by the imaginary unit, denoted as 'i', where \( i^2 = -1 \). This might seem perplexing because we can't find a real number that, when squared, gives a negative result. However, in mathematics, we accept 'i' and its multiples, like \(2i\) from our exercise, as legitimate solutions to certain types of equations.

In practical terms, imaginary numbers are not \