Solving The Polynomial Equation $x^3 + X^2 = -9x - 9$ Finding The Other Root

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In the fascinating world of algebra, polynomial equations hold a special place. These equations, with their variables raised to various powers, can describe a myriad of phenomena, from the trajectory of a ball to the growth of a population. One such equation, $x^3 + x^2 = -9x - 9$, presents us with an intriguing challenge. Our mission is to unravel its secrets and discover all its roots, particularly the elusive 'other root' lurking amidst the complex solutions.

Unveiling the Equation: A Step-by-Step Approach

To begin our exploration, let's first rewrite the equation in its standard form. This involves moving all terms to one side, setting the equation equal to zero. By adding $9x$ and $9$ to both sides, we transform the equation into:

$x^3 + x^2 + 9x + 9 = 0$

Now, we have a cubic polynomial equation, a third-degree equation, staring back at us. Cubic equations can be tricky to solve directly, but we have a valuable piece of information at our disposal: the equation has complex roots $\pm 3i$. This is a crucial clue that will guide us towards the solution.

Leveraging the Complex Roots

Complex roots of polynomials with real coefficients always come in conjugate pairs. This means that if $3i$ is a root, then its conjugate, $-3i$, must also be a root. This property is a cornerstone of complex number theory and provides a powerful tool for solving polynomial equations.

Knowing two roots, $3i$ and $-3i$, we can construct a quadratic factor of the polynomial. A factor is an expression that divides the polynomial evenly, leaving no remainder. Since $3i$ is a root, $(x - 3i)$ must be a factor. Similarly, since $-3i$ is a root, $(x + 3i)$ must also be a factor. Multiplying these two factors together, we get:

(x - 3i)(x + 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9

This quadratic expression, $x^2 + 9$, is a factor of our original cubic polynomial. Now, we can use this information to find the remaining root.

Polynomial Division: Unearthing the Remaining Factor

To find the other factor, we can perform polynomial division. We divide the original polynomial, $x^3 + x^2 + 9x + 9$, by the quadratic factor we just found, $x^2 + 9$. Polynomial division is a systematic way of dividing polynomials, similar to long division with numbers.

Performing the division, we get:

(x^3 + x^2 + 9x + 9) / (x^2 + 9) = x + 1

The result of the division is $x + 1$, which is the remaining factor of the cubic polynomial.

The Final Piece: Determining the Other Root

Now that we have all the factors of the polynomial, we can find the roots. The roots are the values of $x$ that make the polynomial equal to zero. We already know that $x = 3i$ and $x = -3i$ are roots from the complex conjugate pair. To find the remaining root, we set the factor $x + 1$ equal to zero and solve for $x$:

x + 1 = 0

x = -1

Therefore, the other root of the polynomial equation is -1.

Deep Dive into Polynomial Roots and Complex Numbers

To truly appreciate the solution, it's essential to delve deeper into the concepts of polynomial roots and complex numbers. Polynomial roots, also known as zeros, are the values of the variable that make the polynomial expression equal to zero. These roots are fundamental to understanding the behavior and properties of polynomials.

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a cornerstone of polynomial theory. It states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem guarantees the existence of roots for any polynomial equation, though finding those roots can be a challenging task.

Furthermore, a polynomial of degree $n$ (the highest power of the variable) has exactly $n$ complex roots, counting multiplicities. Multiplicity refers to the number of times a root appears as a solution. For example, in the equation $(x - 2)^2 = 0$, the root $x = 2$ has a multiplicity of 2.

In our case, the cubic equation $x^3 + x^2 + 9x + 9 = 0$ has a degree of 3, so it has exactly three complex roots. We found two complex roots, $3i$ and $-3i$, and one real root, $-1$. This aligns perfectly with the Fundamental Theorem of Algebra.

The Significance of Complex Conjugate Pairs

As we encountered earlier, complex roots of polynomials with real coefficients always occur in conjugate pairs. This is a crucial property that simplifies the process of finding roots. A complex conjugate is formed by changing the sign of the imaginary part of a complex number. For example, the complex conjugate of $a + bi$ is $a - bi$.

The reason for this pairing lies in the nature of polynomial coefficients. When a polynomial has real coefficients, the imaginary parts of complex roots must cancel out in the polynomial expression. This cancellation occurs naturally when complex conjugate pairs are present.

In our problem, the presence of the complex root $3i$ immediately implied the existence of its conjugate, $-3i$. This allowed us to construct the quadratic factor $x^2 + 9$, significantly simplifying the solution process.

Visualizing Roots on the Complex Plane

The complex plane provides a visual representation of complex numbers and their relationships. It's a two-dimensional plane where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part.

Each complex number can be plotted as a point on the complex plane. The roots of a polynomial equation can also be visualized on the complex plane. The Fundamental Theorem of Algebra guarantees that a polynomial of degree $n$ will have $n$ roots (counting multiplicities) in the complex plane.

In our example, the roots $3i$ and $-3i$ are located on the imaginary axis, equidistant from the real axis. The root $-1$ is located on the real axis. The visual representation on the complex plane reinforces the concept of complex conjugate pairs and the distribution of roots for a polynomial equation.

Alternative Approaches to Solving the Equation

While we solved the equation using polynomial division and the complex conjugate root theorem, there are alternative methods we could have employed. Exploring these methods provides a broader understanding of polynomial solving techniques.

Factoring by Grouping

Factoring by grouping is a technique that can sometimes be used to factor polynomials, particularly those with four terms. This method involves grouping terms together and factoring out common factors from each group.

Let's apply this technique to our equation, $x^3 + x^2 + 9x + 9 = 0$. We can group the first two terms and the last two terms:

(x^3 + x^2) + (9x + 9) = 0

Now, we factor out the common factors from each group:

x^2(x + 1) + 9(x + 1) = 0

Notice that we now have a common factor of $(x + 1)$ in both terms. We can factor this out:

(x + 1)(x^2 + 9) = 0

This is the same factorization we obtained using polynomial division. We can then proceed to find the roots as before, setting each factor equal to zero.

Using the Rational Root Theorem

The Rational Root Theorem provides a way to identify potential rational roots of a polynomial equation with integer coefficients. A rational root is a root that can be expressed as a fraction $p/q$, where $p$ and $q$ are integers.

The theorem states that if a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0$ has a rational root $p/q$, then $p$ must be a factor of the constant term $a_0$, and $q$ must be a factor of the leading coefficient $a_n$.

In our equation, $x^3 + x^2 + 9x + 9 = 0$, the constant term is 9, and the leading coefficient is 1. Therefore, the possible rational roots are the factors of 9 divided by the factors of 1, which are $\pm 1, \pm 3, \pm 9$.

We can test these potential roots by substituting them into the equation. We find that $x = -1$ is a root, confirming our previous result. However, the Rational Root Theorem only helps us find rational roots; it doesn't reveal complex roots.

Conclusion: The Power of Polynomial Equations

The journey through the polynomial equation $x^3 + x^2 = -9x - 9$ has been a testament to the power and elegance of algebra. By leveraging the complex conjugate root theorem, polynomial division, and factoring techniques, we successfully unearthed all the roots of the equation, including the elusive 'other root,' which turned out to be -1.

This exploration has also highlighted the importance of fundamental concepts like the Fundamental Theorem of Algebra and the significance of complex numbers in polynomial theory. These concepts provide a framework for understanding the behavior and properties of polynomials, enabling us to solve complex problems and unlock hidden mathematical truths.

Polynomial equations are not just abstract mathematical constructs; they are powerful tools for modeling and understanding the world around us. From engineering and physics to economics and computer science, polynomials play a crucial role in solving real-world problems. By mastering the techniques for solving polynomial equations, we gain a valuable skill that can be applied across a wide range of disciplines.

As we conclude our exploration, let us remember that the beauty of mathematics lies not only in the solutions we find but also in the journey of discovery itself. The challenges and insights gained along the way enrich our understanding and appreciation for the intricate patterns and relationships that govern the mathematical universe.