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Find a degree 
Magazine Chapter 3: Problem 36
Find a polynomial with integer coefficients that satisfies the given conditions. \(R\) has degree \(4,\) and zeros \(1-2 i\) and \(1,\) with 1 a zero of multiplicity 2
Short Answer
Expert verified
The polynomial is \(x^4 - 4x^3 + 10x^2 - 12x + 5.\)
Step by step solution
01
Identify the Zeros
The polynomial has zeros at \(1 - 2i\) and \(1\). Since the degree is 4 and one zero is \(1\) with multiplicity 2, we need another zero. By the complex conjugate root theorem, if a polynomial with real coefficients has a complex zero, then its conjugate is also a zero. Thus, the zeros of the polynomial are \(1, 1, 1 - 2i, 1 + 2i\).
02
Construct Factors from Zeros
Each zero \(a\) of a polynomial corresponds to a factor \((x-a)\). Therefore, for the given zeros, the factors are \((x-1), (x-1), (x-(1-2i)), (x-(1+2i))\).
03
Form the Polynomial Using the Factors
Combine the factors to form the polynomial: \[R(x) = (x-1)^2 (x-(1-2i))(x-(1+2i)).\]
04
Multiply the Complex Conjugate Pair
Calculate the product of the complex conjugate factors: \((x-(1-2i))(x-(1+2i)) = ((x-1) + 2i)((x-1) - 2i)\). Using the difference of squares formula, this becomes: \((x-1)^2 - (2i)^2 = (x-1)^2 + 4\) because \(i^2 = -1\). This simplifies to \(x^2 - 2x + 1 + 4 = x^2 - 2x + 5\).
05
Expand the Polynomial Fully
Now multiply the result from Step 4 with the factor \((x-1)^2\): \[(x-1)^2 = (x-1)(x-1) = x^2 - 2x + 1\] Multiply this with \(x^2 - 2x + 5\): \[ (x^2 - 2x + 1)(x^2 - 2x + 5) = x^4 - 2x^3 + 5x^2 - 2x^3 + 4x^2 - 10x + x^2 - 2x + 5.\] Combine like terms to get: \[ x^4 - 4x^3 + 10x^2 - 12x + 5.\]
06
Verify the Solution
The resulting polynomial \(R(x) = x^4 - 4x^3 + 10x^2 - 12x + 5\) is a degree 4 polynomial with integer coefficients, and it satisfies the given conditions of having the specified zeros.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Degree of a Polynomial
Understanding the degree of a polynomial is crucial in solving polynomial equations. The degree refers to the highest power of the variable in a polynomial. For example, in the polynomial \( R(x) = x^4 - 4x^3 + 10x^2 - 12x + 5 \), the degree is 4 because the highest exponent of \( x \) is 4.
The degree tells us several things about the polynomial:
The degree tells us several things about the polynomial:
- It indicates the maximum number of roots or zeros the polynomial can have.
- It affects the general shape of the polynomial's graph.
- It plays a role in determining the end behavior of the polynomial function.
Complex Conjugate Root Theorem
The Complex Conjugate Root Theorem is a fundamental rule when dealing with polynomials with real coefficients. It states that if a polynomial has real coefficients and includes a complex number as a root, its complex conjugate must also be a root.
Let's unpack this with the original problem. We have a root \( 1 - 2i \). According to the theorem, the complex conjugate \( 1 + 2i \) must also be a root. This ensures that the polynomial can be constructed with real coefficients, as complex roots naturally appear in conjugate pairs.
Let's unpack this with the original problem. We have a root \( 1 - 2i \). According to the theorem, the complex conjugate \( 1 + 2i \) must also be a root. This ensures that the polynomial can be constructed with real coefficients, as complex roots naturally appear in conjugate pairs.
- Complex roots appear in pairs: \( (a+bi), (a-bi) \).
- This property stabilizes the polynomial by allowing it to maintain symmetry.
Multiplicity of Zeros
The multiplicity of a zero refers to the number of times a particular root is repeated in a polynomial. A zero with multiplicity 2 means it appears twice as a root. In the exercise, the zero \( 1 \) has a multiplicity of 2, contributing to the total degree of 4.
This concept tells us:
This concept tells us:
- If a root \( r \) has a multiplicity of \( m \), the factor \((x-r)^m\) is present in the polynomial.
- The graph of the polynomial will 'touch and bounce off' the x-axis at this point, rather than crossing it.
- Multiplicities determine the smoothness or flatness at the x-intercepts of the polynomial's graph.
Difference of Squares Formula
The Difference of Squares Formula is a useful tool in simplifying expressions, particularly when dealing with complex conjugates. This formula is expressed as \((a+b)(a-b) = a^2 - b^2\).
In the original solution, it was used to simplify the expression \((x-(1-2i))(x-(1+2i))\). Applying the formula, this becomes \((x-1)^2 - (2i)^2\). Since \( i^2 = -1 \), this simplifies to \((x-1)^2 + 4\).
This step is essential as it allows:
In the original solution, it was used to simplify the expression \((x-(1-2i))(x-(1+2i))\). Applying the formula, this becomes \((x-1)^2 - (2i)^2\). Since \( i^2 = -1 \), this simplifies to \((x-1)^2 + 4\).
This step is essential as it allows:
- The conversion of products of complex conjugates into a polynomial with real coefficients.
- Simplifies the multiplication needed to find the final polynomial expression.
