Finding The Real Root Of A Cubic Polynomial

Let's dive into solving polynomial equations, specifically focusing on how to find all the roots when you already know some of them. This is super useful, especially when dealing with complex roots. We'll break down a specific example step by step, making sure you understand the logic behind each move.

Understanding the Problem

We are given the polynomial equation $x^3 + x^2 = -9x - 9$, and we know that $\pm 3i$ are complex roots. Our mission is to find the remaining root. First, let's rewrite the equation in the standard polynomial form:

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

Now, the key here is understanding that complex roots of polynomials with real coefficients always come in conjugate pairs. That means if $3i$ is a root, then its conjugate, $-3i$, must also be a root. This is a crucial piece of information because it lets us factor the polynomial.

Using Known Roots to Factor

Since we know two roots, $3i$ and $-3i$, we can determine a quadratic factor of the polynomial. Remember that if $r$ is a root of a polynomial, then $(x - r)$ is a factor. So, we have the factors $(x - 3i)$ and $(x + 3i)$. Let's multiply these together:

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

This tells us that $x^2 + 9$ is a factor of our cubic polynomial $x^3 + x^2 + 9x + 9$. Now, we need to find the other factor. To do this, we'll perform polynomial division.

Polynomial Division

We're going to divide $x^3 + x^2 + 9x + 9$ by $x^2 + 9$. Here’s how polynomial long division works:

 x + 1
------------------
x^2+9 | x^3 + x^2 + 9x + 9
       - (x^3 + 0x^2 + 9x)
------------------
             x^2 + 0x + 9
             - (x^2 + 0x + 9)
------------------
                     0

The result of the division is $x + 1$. This means that our original polynomial can be factored as:

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

Finding the Remaining Root

Now that we've factored the polynomial, finding the remaining root is easy. We already know the roots from the factor $(x^2 + 9)$ are $\pm 3i$. So, the remaining root must come from the factor $(x + 1)$. To find this root, we set the factor equal to zero:

$x + 1 = 0$

Solving for $x$, we get:

$x = -1$

So, the other root is $-1$.

Conclusion

Therefore, the polynomial equation $x^3 + x^2 = -9x - 9$ has complex roots $\pm 3i$, and the other root is $-1$. The correct answer is B. $-1$.

Knowing that complex roots come in conjugate pairs and using polynomial division are powerful tools for solving these types of problems. Keep practicing, and you'll become a pro at finding roots of polynomials!

Additional Insights and Tips

The Conjugate Root Theorem

As we've seen, the Conjugate Root Theorem is essential. It states that if a polynomial with real coefficients has a complex root $a + bi$, then its complex conjugate $a - bi$ is also a root. This theorem helps us simplify the problem because we instantly know another root when one complex root is given. This is especially useful in problems like this where we need to find all the roots of the polynomial.

Factoring and Polynomial Division

Factoring polynomials is a fundamental skill in algebra. When some roots are known, factoring becomes more manageable. We can construct factors corresponding to the known roots and then use polynomial division to find the remaining factors. It’s like reverse engineering! The polynomial division process helps us break down a higher-degree polynomial into simpler, manageable components.

Synthetic Division

For those comfortable with synthetic division, it can be a faster alternative to polynomial long division, especially when dividing by a linear factor $(x - r)$. However, in this case, since we were dividing by a quadratic $x^2 + 9$, polynomial long division was more appropriate.

Verifying the Roots

Always verify your roots by plugging them back into the original equation to ensure they satisfy the equation. This is a simple yet effective way to catch any mistakes made during the factoring or division process. For instance, plugging $x = -1$ back into the equation:

$(-1)^3 + (-1)^2 = -9(-1) - 9$

$-1 + 1 = 9 - 9$

$0 = 0$

This confirms that $x = -1$ is indeed a root.

Common Mistakes to Avoid

• Forgetting the Conjugate: A common mistake is to forget that complex roots come in conjugate pairs. Always consider the conjugate when given a complex root.
• Incorrect Division: Polynomial division can be tricky. Make sure to align terms correctly and pay close attention to signs.
• Not Verifying: Failing to verify the roots can lead to incorrect answers. Always double-check your work.

Practice Makes Perfect

The more you practice these types of problems, the more comfortable you'll become with the techniques. Try solving similar problems with different polynomials and roots. Focus on understanding the underlying principles, and you'll be well on your way to mastering polynomial equations.

Real-World Applications

While polynomial equations might seem abstract, they have many real-world applications, particularly in engineering, physics, and computer science. They are used to model various phenomena, from the trajectory of a projectile to the design of circuits. Understanding how to solve these equations is crucial for solving complex problems in these fields.

The Rational Root Theorem

If you're struggling to find the initial roots, the Rational Root Theorem can be helpful. It provides a list of possible rational roots based on the coefficients of the polynomial. Although it doesn't guarantee a rational root exists, it can be a good starting point for finding potential roots to test.

By mastering these techniques and understanding the underlying principles, you can confidently tackle polynomial equations and find all their roots. Good luck, and happy solving!