Problem 19 - Algebra for College Students

Question

Find a polynomial equation with real coefficients that has the given roots. $$-2 i, 2 i$$

Step-by-Step Solution

Answer

The polynomial is \(x^2 + 4\).

Step 1: Identify the roots

The given roots are \(-2i\) and \(2i\). These roots are complex conjugates of each other because they have the same magnitude but opposite signs.

Step 2: Use the fact that complex conjugate roots appear in pairs

For a polynomial with real coefficients, if \(2i\) is a root, then \(-2i\) must also be a root because complex roots always come in conjugate pairs when coefficients are real.

Step 3: Set up the polynomial factors

Roots of a polynomial can be turned into factors. For the roots \(-2i\) and \(2i\), the corresponding factors are \((x - (-2i))\) and \((x - 2i)\). This simplifies to \((x + 2i)\) and \((x - 2i)\).

Step 4: Multiply the factors to form the polynomial

Expand the product of the factors \((x + 2i)(x - 2i)\). Using the difference of squares formula: \[(x + 2i)(x - 2i) = x^2 - (2i)^2\]Since \(i^2 = -1\), \((2i)^2 = -4\). Therefore,\[x^2 - (-4) = x^2 + 4\]

Step 5: Write the final polynomial

The expanded form gives the polynomial \(x^2 + 4\). This is the polynomial with real coefficients that has the given roots \(-2i\) and \(2i\).

Key Concepts

Complex Roots

In the world of polynomials, sometimes we encounter complex numbers as roots. A complex number has two parts: a real part and an imaginary part. Imaginary parts are multiples of the imaginary unit, represented as 'i', where \(i^2 = -1\). For instance, in the exercise above, we work with the roots -2i and 2i. These numbers do not appear on the real number line but rather in the complex plane. One characteristic of complex roots in polynomials with real coefficients is that they always come in pairs known as 'complex conjugates'. Complex conjugates have the same real part but opposite imaginary parts. So, if 2i is a root, -2i also must be a root.

Real Coefficients

A polynomial equation is simply an expression consisting of variables, coefficients, and exponents, summed together. When we say a polynomial has 'real coefficients', we mean that all the numbers multiplying the variables in the polynomial are real numbers (not imaginary). In the provided exercise, you see that we needed to form a polynomial with real coefficients. If a polynomial includes complex roots, these roots must appear in conjugate pairs to ensure that the coefficients of the polynomial remain real. This is important because the presence of a lone complex root would result in coefficients that are not purely real numbers.

Difference of Squares

During the multiplication of polynomial factors, we often encounter the 'difference of squares' formula. This is a simple and powerful algebraic identity that helps simplify expressions like the one above. The formula states that:
\((a + b)(a - b) = a^2 - b^2\)
In our case, the 'a' is x and the 'b' is 2i. When we plug in these values, we get:
\( (x + 2i)(x - 2i) = x^2 - (2i)^2\).
Remember that \(i^2 = -1\), so \( (2i)^2 = -4\). Thus, our expression simplifies to:
\(x^2 - (-4) = x^2 + 4\).
Using the difference of squares formula is crucial for transforming polynomial factors into a simpler, expanded form. This form helps us ensure that the polynomial equation retains real coefficients.