91Ó°ÊÓ

In the realm of polynomial equations, complex roots often enter the scene. Complex numbers extend our traditional set of real numbers, adding in elements of the form \(a + bi\), where \(i\) represents the imaginary unit. This unit is defined by the property that \(i^2 = -1\). Complex numbers are crucial because they ensure that any polynomial equation—no matter its degree—has a solution.
When dealing with polynomial equations with real coefficients, you must remember that complex roots always come in pairs known as 'conjugate pairs'. Understanding this will help you accurately construct polynomials from given roots.

When a polynomial has real coefficients, any complex root must have a conjugate. A conjugate pair is made up of two complex numbers of the form \(a + bi\) and \(a - bi\). This is because the sum and product of conjugate pairs are both real numbers. Let's look at the roots in this exercise:

• Root 1: \(-9i\)
• Root 2: \(9i\)

These roots are purely imaginary and indeed form a conjugate pair because their imaginary parts are equal in magnitude but opposite in sign.
Constructing the polynomial from these roots requires creating factors from each root and then multiplying them together. For the roots \(-9i\) and \(9i\), the factors are \((x + 9i)\) and \((x - 9i)\).Multiplying these factors yields a polynomial with real coefficients, thanks to an important algebraic identity: the difference of squares.

The difference of squares formula is a handy tool when you're dealing with conjugate pairs. This algebraic identity is expressed as
\[a^2 - b^2 = (a - b)(a + b)\]
In our context, for the roots \(-9i\) and \(9i\), we use
\[(x + 9i)(x - 9i) = x^2 - (9i)^2\]
The imaginary unit \(i\) satisfies \(i^2 = -1\). Applying this to \(9i\), we have
\[9^2*i^2 = 81*(-1) = -81\]

Thus,

• The factors \((x + 9i)\) and \((x - 9i)\) multiply to form \[x^2 - (9i)^2 = x^2 - (-81) = x^2 + 81\]

This resulting polynomial, \(x^2 + 81\), has real coefficients and the given complex roots. Mastering these concepts provides a strong foundation in understanding higher-level algebra and the behavior of polynomial equations.