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Find a degree 
Magazine Chapter 3: Problem 24
Find a polynomial with real coefficients satisfying the given conditions. Find a polynomial of lowest degree with real coefficients and the given zeros. $$x=-1 \text { and } x=1-i$$
Short Answer
Expert verified
The polynomial is \(x^3 - x^2 + 2\).
Step by step solution
01
Identify the Zeros and Their Conjugates
The problem states that the polynomial has zeros at \(x = -1\) and \(x = 1 - i\). Since the polynomial needs to have real coefficients, the complex zero \(x = 1 - i\) requires a conjugate zero \(x = 1 + i\) as well. So, the zeros of the polynomial are \(x = -1\), \(x = 1 - i\), and \(x = 1 + i\).
02
Write Factors for Each Zero
Express the polynomial as a product of factors corresponding to each zero. The factors will be \((x + 1)\), \((x - (1 - i))\), and \((x - (1 + i))\).
03
Simplify the Complex Factors
Multiply the complex conjugate factors to eliminate imaginary parts: \[(x - (1 - i))(x - (1 + i)) = (x - 1 + i)(x - 1 - i).\] Use the identity \((a + b)(a - b) = a^2 - b^2\). Here, \(a = x - 1\) and \(b = i\), so the product simplifies to: \[(x - 1)^2 - i^2 = (x - 1)^2 - (-1) = (x - 1)^2 + 1.\]
04
Expand the Binomial
Expand \((x - 1)^2 + 1\): \[(x - 1)^2 = x^2 - 2x + 1.\] Therefore, \[(x - 1)^2 + 1 = x^2 - 2x + 1 + 1 = x^2 - 2x + 2.\]
05
Form the Polynomial Expression
Multiply the factor for the real zero \(x + 1\) by the simplified complex expression: \[(x + 1)(x^2 - 2x + 2).\]
06
Expand the Expression
Expand the product to find the polynomial: \[(x + 1)(x^2 - 2x + 2) = x(x^2 - 2x + 2) + 1(x^2 - 2x + 2). \]\[x(x^2 - 2x + 2) = x^3 - 2x^2 + 2x. \]\[1(x^2 - 2x + 2) = x^2 - 2x + 2.\]Combine these to get: \[x^3 - 2x^2 + 2x + x^2 - 2x + 2 = x^3 - x^2 + 2.\]
07
Final Step: Present the Polynomial
The polynomial of lowest degree with the given zeros and real coefficients is \[x^3 - x^2 + 2.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Conjugate Roots
When working with polynomials that have real coefficients, there's a fascinating rule regarding complex numbers. If a complex number is a root (or zero) of the polynomial, then its complex conjugate must also be a root. This means that if we have a root like \(1-i\), the polynomial must also include \(1+i\) as a root.
- Complex numbers are numbers that have a real part and an imaginary part, formatted as \(a + bi\), where \(i\) is the imaginary unit \(\sqrt{-1}\).
- Complex conjugates are pairs of complex numbers that have the same real component but opposite imaginary components. For example, \(1-i\) and \(1+i\).
Polynomial Expansion
Polynomial expansion involves expressing a product of factors as a sum. This typically comes into play after identifying the factors related to each root of a polynomial. After capturing the zeros of the polynomial as factors, the next step is to expand these factors into a full polynomial expression. For example, if we have the factors
- \((x+1)\) for the zero \(-1\),
- \((x-1+i)(x-1-i)\) to account for zeros \(1-i\) and \(1+i\).
Binomial Theorem
The binomial theorem is a powerful way to expand expressions that are raised to a power. This theorem helps in breaking down expressions like \((x - 1)^2\) that arise in polynomial problems.
- The formula is \((x-y)^2 = x^2 - 2xy + y^2\).
- It lets us expand binomials into manageable polynomial terms.
Degree of a Polynomial
The degree of a polynomial is a fundamental concept. It indicates the polynomial鈥檚 highest power of the variable. This is crucial because it determines the polynomial's behavior and complexity.
- A polynomial's degree corresponds to the number of roots it can have, including complex roots.
- The polynomial in our problem has a degree of 3, coming from the factor \((x + 1)\) and the quadratic result of the complex conjugates \((x^2 - 2x + 2)\).
