/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Suppose that a polynomial functi... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 24

Suppose that a polynomial function of degree 4 with rational coefficients has the given numbers as zeros. Find the other zero( \(s\) ). $$6-5 i,-1+\sqrt{7}$$

Short Answer

Expert verified
The polynomial's zeros are 6 - 5i, 6 + 5i, -1 + \( \sqrt{7} \), -1 - \( \sqrt{7} \).

Step by step solution

01

Identify known zeros and their properties

Given zeros are complex numbers with imaginary parts: 6 - 5i and -1 + \( \sqrt{7} \). Since the coefficients are rational, their complex conjugates will also be zeros: 6 + 5i and -1 - \( \sqrt{7} \).
02

List known zeros and their conjugates

The zeros listed are: \(6 - 5i, 6 + 5i, -1 + \( \sqrt{7} \), -1 - \( \sqrt{7} \) \). Since a degree 4 polynomial can have up to 4 zeros, these are all four zeros of the polynomial.

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Key Concepts

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

Degree of Polynomial
A polynomial's degree is the highest power of the variable in the polynomial. If a polynomial is described as having a degree of 4, the highest exponent on the variable, typically denoted as 饾懃, is 4. The degree directly influences the number of roots or zeros the polynomial can have. For instance, a polynomial of degree 4 can have up to four zeros. These zeros might be real or complex, and they might be distinct or repeated. In our exercise, the polynomial of degree 4 uses this property, allowing us to aim for four zeros.
Complex Conjugate
In mathematics, every complex number has a twin called its complex conjugate. If a complex number is written as 饾憥 + 饾憦饾憱, where 饾憥 and 饾憦 are real numbers and 饾憱 is the imaginary unit, its complex conjugate is 饾憥 - 饾憦饾憱. When a polynomial has rational coefficients, if it has any complex zeros, their conjugates must also be zeros. This symmetry ensures the polynomial remains real-valued when evaluated with rational numbers. In our case, the known zeros include complex numbers: 6 - 5饾憱 and -1 + 鈭�7. Their complex conjugate pairs, 6 + 5饾憱 and -1 - 鈭�7, also need to be included as zeros.
Rational Coefficients
Polynomials with rational coefficients have a special relationship with their complex roots. If a polynomial equation has rational coefficients, then complex roots must appear in conjugate pairs to keep the coefficients rational. Rational coefficients are coefficients that can be expressed as fractions of integers, including whole numbers. In our exercise, rational coefficients mean that complex zeros will always come with their complex conjugates. For the polynomial with degree 4 provided, having rational coefficients requires the pairings 6 - 5饾憱 with 6 + 5饾憱, and -1 + 鈭�7 with -1 - 鈭�7, making sure no irrational parts remain isolated, ensuring all coefficients are rational.

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