Problem 41 - Algebra and Trigonometry

Question

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 \(2 .\)

Step-by-Step Solution

Answer

The polynomial is \(x^4 - 4x^3 + 10x^2 - 12x + 5\).

Step 1: Understanding the Roots

The polynomial has a degree of 4 and the following zeros: 1 with multiplicity 2 and \(1-2i\). Since polynomials with real coefficients must have complex roots in conjugate pairs, \(1+2i\) must also be a zero of the polynomial.

Step 2: Constructing Factors

Each zero corresponds to a factor of the polynomial. Thus, 1 (multiplicity 2) corresponds to the factor \((x-1)^2\), and the complex conjugate zeros \(1-2i\) and \(1+2i\) correspond to the factors \((x - (1-2i))\) and \((x -(1+2i))\).

Step 3: Combining Complex Factors

Multiply the factors \((x - (1-2i))\) and \((x -(1+2i))\) to form a real polynomial part. The product is calculated as: \[(x - (1-2i))(x -(1+2i)) = ((x-1) + 2i)((x-1) - 2i) = [(x-1)^2 - (2i)^2] = (x-1)^2 + 4.\]

Step 4: Expanding Polynomial

The polynomial can be expressed as \[R(x) = (x-1)^2[(x-1)^2 + 4].\]First, expand \((x-1)^2\): \[ (x-1)^2 = x^2 - 2x + 1.\]Then, substitute and expand:\[R(x) = (x^2 - 2x + 1)(x^2 - 2x + 5).\]

Step 5: Final Expansion

Expand the polynomial \[(x^2 - 2x + 1)(x^2 - 2x + 5)\] through distribution:\[x^2(x^2 - 2x + 5) - 2x(x^2 - 2x + 5) + 1(x^2 - 2x + 5)\].Expand each term:\[x^4 - 2x^3 + 5x^2 - 2x^3 + 4x^2 - 10x + x^2 - 2x + 5.\]Combine like terms:\[x^4 - 4x^3 + 10x^2 - 12x + 5.\]

Step 6: Verification

Check the degree and the zeros. The polynomial derived \(R(x) = x^4 - 4x^3 + 10x^2 - 12x + 5\) has degree 4 and satisfies the zero conditions: 1 with multiplicity 2, \(1-2i\), and \(1+2i\) are all accounted for.

Key Concepts

Polynomial Degree

Polynomials are an important aspect of algebra and understanding their degree can be very helpful. The degree of a polynomial is the highest power of the variable, usually represented as 'x', in the polynomial. For example, the polynomial \(x^4 - 4x^3 + 10x^2 - 12x + 5\) is a degree 4 polynomial because the highest exponent of 'x' is 4.

The degree provides information on the behavior of the polynomial, especially regarding its roots and the number of times it intersects the x-axis (real roots). For instance, a polynomial of degree \(n\) can have at most \(n\) roots. Each root corresponds to an intersection on the graph, therefore, the polynomial in our problem is expected to have up to 4 roots.

In summary, knowing the degree of a polynomial helps us identify its overall structure.

Complex Conjugate Roots

Understanding complex conjugate roots is crucial for working with polynomials that have complex numbers as zeros. While dealing with polynomials that have real coefficients, any complex roots must occur in conjugate pairs. This means if \(a + bi\) is a root, then \(a - bi\) must also be a root.

In the example exercise, we have the root \(1 - 2i\). According to the rule of conjugate pairs, \(1 + 2i\) must also be a root to maintain the integrity of the polynomial with integer coefficients.

• These conjugate pairs ensure that when you multiply the factors corresponding to these roots, the imaginary parts cancel out.
• This results in a quadratic with real coefficients.

This characteristic of complex roots makes them interesting and essential when simplifying polynomials.

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular solution is repeated for a polynomial equation. If a polynomial has a zero with a multiplicity greater than one, it means the polynomial will touch or "bounce off" the x-axis at that point without crossing it.

In the specified problem, the zero 1 has a multiplicity of 2. This means the corresponding factor in the polynomial appears twice. Therefore, the factor associated with this zero is \((x - 1)^2\).

• When multiplied out, this factor affects the shape and position of the polynomial's graph.
• A zero with even multiplicity such as 2 causes the graph to touch the x-axis and turn back.

Understanding multiplicity is key in determining the polynomial's behavior at its x-intercepts.

Factorization of Polynomials

Factorization is the process of breaking down a polynomial into a product of simpler polynomials, which when multiplied together, give the original polynomial. This process makes it easier to solve polynomial equations.

The exercise involves constructing a polynomial from given roots by turning each root into a corresponding factor. For example, the roots \(1, 1, 1-2i, 1+2i\) transformed into factors form \((x-1)^2(x-(1-2i))(x-(1+2i))\).

• The complex factors \((x-(1-2i))(x-(1+2i))\) can be multiplied together to form a quadratic polynomial without imaginary parts.
• The polynomial can therefore be expressed and expanded into a product of these factors, yielding a polynomial with real and integer coefficients.

Factorization simplifies the polynomial and allows us to easily re-construct the polynomial if needed.