Problem 5 - Algebra and Trigonometry

Question

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{4}+4 x^{2}\)

Step-by-Step Solution

Answer

Zeros: 0, 2i, -2i. Factorized: \(P(x) = x^2(x - 2i)(x + 2i)\).

Step 1: Identify Common Factor

First, observe that each term in the polynomial has a common factor. In this polynomial, the common factor is \(x^2\). We can factor out \(x^2\) from the polynomial: \[P(x) = x^2(x^2 + 4)\].

Step 2: Find Zeros of Factored Terms

Set each factor of the polynomial equal to zero to find the zeros. The first factor \(x^2 = 0\) gives a zero at \(x = 0\).

Step 3: Solve for Zeros in the Quadratic Expression

Now, solve the equation derived from the quadratic expression: \(x^2 + 4 = 0\). Rearrange it to find \(x^2 = -4\).

Step 4: Calculate Complex Zeros

To solve \(x^2 = -4\), take the square root of both sides: \(x = \pm \sqrt{-4}\). Using the imaginary unit \(i\), simplify this to \(x = \pm 2i\).

Step 5: List All Zeros

Combine all zeros found from each factor. The polynomial \(P(x) = x^4 + 4x^2\) has zeros at \(x = 0, x = 2i, x = -2i\), where \(0\) is a zero of multiplicity two.

Step 6: Write the Complete Factorization

Now that all zeros are known, express the polynomial in factored form using these zeros: \[P(x) = x^2(x - 2i)(x + 2i)\].

Key Concepts

Complex Numbers

Complex numbers are essential whenever we encounter square roots of negative numbers. In mathematics, a complex number is expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit that satisfies \(i^2 = -1\). These numbers enable us to solve equations that do not have real solutions, such as \(x^2 + 1 = 0\).

When tackling a polynomial that produces negative values inside a square root, complex numbers provide a solution path. In the polynomial from the exercise, the zero derived from \(x^2 = -4\) was not a real number. Instead, it solved to the complex numbers \(x = 2i\) and \(x = -2i\). This showcases how complex numbers extend our number system, making it possible to factor polynomials completely into linear factors.

Zeros of a Polynomial

Finding the zeros of a polynomial is a critical step in understanding its behavior and factoring it. A zero of a polynomial is a solution to the equation \(P(x) = 0\). These can be either real or complex numbers. In the polynomial \(P(x) = x^4 + 4x^2\), zeros were found using the factored form \(x^2(x^2 + 4)\).

Steps to find zeros:

• Factor the polynomial to identify simpler expressions.
• Set each expression equal to zero.
• Solve for the values of \(x\) that satisfy each equation.

The zeros found were \(x = 0\) with a multiplicity of two, and \(x = 2i\) and \(x = -2i\). These zeros help in the complete factorization of the polynomial, indicating all points where the polynomial crosses or touches the x-axis in the complex plane.

Quadratic Equations

Quadratic equations are polynomials of degree two, typically written as \(ax^2 + bx + c = 0\). The polynomial expression \(x^2 + 4\) from the exercise is a quadratic equation simplified to solve for its zeros. Given no linear term \(b\) and a constant term \(c\), the method to solve such equations usually involves rearranging, completing the square, or using the quadratic formula.

To solve \(x^2 + 4 = 0\), it is rearranged to \(x^2 = -4\), and the square root is taken on both sides. This reveals the zeros as \(x = \pm 2i\).

Important methods for solving quadratic equations:

• Factoring, when possible, to split the squared terms into linear factors.
• Completing the square, which turns the equation into a perfect square trinomial.
• Quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), applicable to any quadratic.

These methods provide systematic approaches to finding zeros for quadratic components within complex polynomials.