Problem 80 - Algebra A Combined Function

Question

Write a polynomial equation that has three solutions: \(0,2 i,\) and \(-2 i\)

Step-by-Step Solution

Answer

The polynomial equation with the solutions 0, 2i, and -2i is \(x^3 + 4x = 0\).

Step 1: Convert Solutions into Factors

The solutions of the polynomial equation are given as \(0, 2i,\) and \(-2i\). Each solution corresponds to a factor of the polynomial. If \(x = 0\) is a solution, the related factor is \(x\). If \(x = 2i\) or \(x = -2i\) is a solution, the related factors are \(x - 2i\) and \(x + 2i\), respectively. Therefore, the factors are \(x, (x - 2i),\) and \(x + 2i\).

Step 2: Multiply the Imaginary Factors

The imaginary factors \((x - 2i)\) and \((x + 2i)\) can be multiplied using the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = x\) and \(b = 2i\), so we have: \((x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 - 4(-1) = x^2 + 4\).

Step 3: Form the Complete Polynomial

Now that we have multiplied the imaginary factors to get \(x^2 + 4\), we can form the complete polynomial by multiplying it by the remaining factor \(x\) from the first step. The complete polynomial is \(x \cdot (x^2 + 4) = x^3 + 4x\).

Step 4: Check the Polynomial

Verify that the polynomial \(x^3 + 4x\) has the solutions \(0, 2i,\) and \(-2i\). The polynomial can be factored back into \(x(x - 2i)(x + 2i)\), confirming that it indeed has the given solutions.

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics. They expand the idea of numbers beyond the real number line. A complex number is typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The unit \(i\) is the imaginary unit, defined by the property \(i^2 = -1\).

Complex numbers allow us to solve equations that don't have real solutions. For example, the equation \(x^2 + 1 = 0\) does not have any real number solutions since no real number squares to a negative value. However, it has complex solutions \(x = i\) and \(x = -i\).

• The addition of complex numbers, \((a + bi) + (c + di)\), is straightforward: \((a+c) + (b+d)i\).
• The multiplication follows the distributive property: \((a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i\).

Learning to work with complex numbers is essential for solving polynomial equations, especially when they result in imaginary solutions.

Imaginary Numbers

Imaginary numbers are a subset of complex numbers. They arise when we attempt to take the square root of a negative number. The basic unit of imaginary numbers is \(i\), which is defined such that \(i^2 = -1\).

Though some may find it abstract, imaginary numbers are important for many applications:

• They are used in engineering, electromagnetism, and signal processing.
• Imaginary numbers appear naturally in contexts like the Fourier transform, simplifying many real-world problems.

The main property to remember about imaginary numbers is their relationship with real numbers, forming complex numbers. When multiplying, remember that \(i^2\) becomes \(-1\), allowing heavier computations to take place by removing the imaginary aspect in certain equations, as evident in solving polynomial equations.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This is a crucial skill in algebra that helps in solving polynomial equations.

Consider a polynomial equation with known solutions. Each solution \(x = r\) translates into a factor of the polynomial, \(x - r\). For example, if the solutions are \(x = 0\), \(x = 2i\), and \(x = -2i\), the factors of the polynomial are \(x\), \(x - 2i\), and \(x + 2i\).

Using the difference of squares formula, \((x - a)(x + a) = x^2 - a^2\), simplifies multiplying these factors:

• Multiply the imaginary factors: \((x - 2i)(x + 2i) = x^2 + 4\).
• Combine with the real factor to form the polynomial: \(x(x^2 + 4) = x^3 + 4x\).

Factoring aids in understanding the structure of a polynomial and provides insight on its solutions. Whether they are real or complex, it's a powerful algebraic tool.