Problem 37 - Algebra and Trigonometry

Question

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ \begin{array}{l}{R \text { has degree } 4, \text { and zeros } 1-2 i \text { and } 1, \text { with } 1 \text { a zero of }} \\ {\text { multiplicity } 2 .}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Zeros and Their Properties

Given zeros of the polynomial are \(1-2i\) and \(1\) with a multiplicity of 2. Since coefficients are integers, the complex conjugate \(1+2i\) must also be a zero due to the Conjugate Root Theorem, meaning the zeros are \(1-2i, 1+2i, 1\) (twice, because of multiplicity).

Step 2: Form Linear Factors from Zeros

Formulate the polynomial's factors using its zeros: \((x - (1 - 2i)), (x - (1 + 2i)), (x - 1)\) with the factor \((x - 1)\) repeated because of its multiplicity. The polynomial is thus expressed as \( (x - (1 - 2i))(x - (1 + 2i))(x - 1)^2 \).

Step 3: Simplify the Complex Conjugate Factors

Multiply the complex conjugate factors: \((x - (1-2i))(x - (1+2i))) = ((x - 1) + 2i)((x - 1) - 2i) = (x - 1)^2 - (2i)^2 = (x - 1)^2 + 4 = x^2 - 2x + 1 + 4 = x^2 - 2x + 5\).

Step 4: Expand the Polynomial

Now, multiply the simplified factor \((x^2 - 2x + 5)\) by the remaining factor \((x - 1)^2 = (x - 1)(x - 1) = x^2 - 2x + 1\).

Step 5: Perform Polynomial Multiplication

Multiply \((x^2 - 2x + 5)\) by \((x^2 - 2x + 1)\):\[(x^2 - 2x + 5)(x^2 - 2x + 1) = x^4 - 2x^3 + x^2 - 2x^3 + 4x^2 - 2x + 5x^2 - 10x + 5\].

Step 6: Combine Like Terms

Combine like terms from the expanded polynomial: \[ x^4 - 4x^3 + 10x^2 - 12x + 5 \]. This is the polynomial with integer coefficients and the given properties.

Key Concepts

Degree 4 Polynomial

When a polynomial is described as having a degree of 4, it means the highest power of the variable, typically denoted as \(x\), is 4. Understanding the degree is crucial in determining the general shape and behavior of the polynomial graph. For instance, a degree 4 polynomial can have:

• Up to four real or complex zeros.
• Three turning points, where the graph changes direction.
• The end behavior depending on the leading coefficient (the coefficient of the term with the highest degree).

The form appears as: \(ax^4 + bx^3 + cx^2 + dx + e\), where \(a, b, c, d,\) and \(e\) are constants, and \(a eq 0\). This structure helps predict the movement of the graph as \(x\) approaches negative or positive infinity. A degree 4 polynomial often fits into categories such as quartic functions and is characterized by a parabolic-like shape but with more complexity.

Zeros of a Polynomial

The zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. In simple terms, they are the intersection points where the graph touches or crosses the x-axis. Understanding the zeros is essential as they provide insights into the solutions and real-world applications of the polynomial.

• Each zero can be real or complex.
• The number of zeros equals the degree of the polynomial, considering multiplicities.
• Zeros can help us factor the polynomial since they are the solutions of the polynomial equation \(P(x) = 0\).

In this exercise, given zeros include \(1-2i\) and \(1\). The complex zeros always appear in conjugate pairs, a fact used in conjunction with the Conjugate Root Theorem to determine all zeros. Counting multiplicities ensures that if a zero is repeated, it is acknowledged in solving and factoring polynomial equations.

Conjugate Root Theorem

The Conjugate Root Theorem is a fundamental concept when dealing with polynomials with real coefficients. It states that if a polynomial has real coefficients and a complex zero, its conjugate must also be a zero. This theorem ensures that complex roots appear in pairs and allows us to maintain polynomial integrity.

• Complex zeros \(a + bi\) inevitably mean \(a - bi\) is also a zero, and vice-versa.
• This principle helps factor polynomials more accurately and symmetrically.
• Ensures that a polynomial with real coefficients cannot have an odd number of complex zeros.

For the polynomial in question, the zero \(1 - 2i\) dictates that \(1 + 2i\) must also be a zero. By applying this theorem, one can ensure the created polynomial adheres to real coefficient rules while maintaining mathematical balance across the complex plane.

Multiplicity of Roots

Multiplicity of roots refers to the number of times a particular zero appears in a polynomial. When a zero has a multiplicity greater than 1, it does not cut through the x-axis but rather touches and returns, indicating repeated roots.

• Affect the shape of the polynomial's graph, often causing horizontal tangencies.
• Multiplicity is an exponent on a factor in the polynomial's factored form.
• Even multiplicity results in the graph touching but not cutting through the axis, while odd multiplicity means crossing the axis.

In our exercise, the zero \(1\) has a multiplicity of 2, indicating it's a double root. It means the polynomial's factorization contains \((x - 1)^2\). This repeated root affects both the way the polynomial is expanded and the overall shape of its graph, giving it specific geometric behavior as it encounters the x-axis.