Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-10: A polynomial \(f(x)\) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express \(f(x)\) as a product of linear and quadratic polynomials with real coefficients that are irreducible over \(\mathbb{R}\). $$ -1,0,3+i ; \quad \text { degree } 4 $$

Step-by-Step Solution

Answer

The polynomial is \(f(x) = x(x + 1)((x - 3)^2 + 1)\).

Step 1: Identifying given information

We are given that the polynomial has zeros \(-1, 0,\) and \(3+i\) and has a degree of 4. Since the coefficients are real, the complex zero \(3+i\) implies that its conjugate \(3-i\) is also a zero.

Step 2: Forming linear factors for real roots

The zeros \(-1\) and \(0\) correspond to the linear factors \((x + 1)\) and \(x\) respectively. These are factors of the polynomial.

Step 3: Creating a quadratic factor for complex roots

The zeros \(3+i\) and \(3-i\) are combined to form the quadratic factor \((x - (3+i))(x - (3-i))\). Use the identity \((a + b)(a - b) = a^2 - b^2\) to simplify this to \((x-3)^2 + 1\).

Step 4: Expressing the polynomial as a product

Combine the linear and quadratic factors to form the polynomial: \(f(x) = x(x + 1)((x - 3)^2 + 1)\).

Step 5: Verifying the polynomial degree

The polynomial \(f(x) = x(x + 1)((x - 3)^2 + 1)\) consists of a degree 1 factor \(x\), a degree 1 factor \((x+1)\), and a degree 2 factor \((x-3)^2 + 1\). The total degree is \(1+1+2=4\), confirming the degree is correct as per the problem statement.

Key Concepts

Complex Roots

When dealing with polynomials that have real coefficients, complex roots always come in conjugate pairs. This means if a polynomial has a complex root such as \(3+i\), it must also have its conjugate \(3-i\) as a root. A complex number \(a+bi\) includes a real part \(a\) and an imaginary part \(b\). The conjugate of this number is \(a-bi\). This ensures the polynomial remains with real coefficients when multiplied out. In the context of polynomial functions, these pairs help ensure that when the polynomial is expressed in real terms, it satisfies the properties needed for it to be entirely real. This is crucial when forming quadratic factors from these roots.

Quadratic Polynomials

Quadratic polynomials are polynomials of degree 2. They are typically expressed in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(aeq0\). In the context of complex roots, when you have a pair of complex conjugates as roots, you can form a quadratic polynomial. For example, given roots \(3+i\) and \(3-i\), they can be multiplied together to form the quadratic \((x-(3+i))(x-(3-i))\). Using the formula \((a+b)(a-b) = a^2-b^2\), this simplifies to \((x-3)^2 + 1\). This quadratic cannot be factored further using real numbers and hence is called irreducible over \(\mathbb{R}\). Such quadratic polynomials are important as they form the building blocks in constructing polynomials of higher degrees.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable in the polynomial expression. It is a key feature that defines the polynomial's behavior and symmetry. For example, the polynomial \(x(x+1)((x-3)^2+1)\) is composed of factors with degrees 1, 1, and 2 respectively, giving it an overall degree of 4. This degree dictates the polynomial's shape, the number of roots it has (including complex roots counted in pairs), and its general growth rate at infinity. Understanding the degree is crucial for analyzing the behavior of polynomials, especially when determining the number and types of roots.

Real Coefficients

A polynomial with real coefficients means all the coefficients in the polynomial are real numbers. This ensures that when complex roots are present, they must appear in conjugate pairs to maintain the polynomial's real nature. For instance, in the polynomial function defined by \(f(x) = x(x+1)((x-3)^2+1)\), each term is crafted such that all coefficients remain real. This can be seen in the quadratic factor \((x-3)^2+1\) derived from the complex roots \(3+i\) and \(3-i\). The condition of real coefficients influences the process of multiplying factors and combining terms to ensure the final polynomial is fit for practical and theoretical applications when solely dealing with real numbers.