Finding Polynomial Zeros Using The Rational Zeros Theorem

Hey guys! Let's dive into the exciting world of polynomials and learn how to find their zeros. In this article, we'll tackle the polynomial function $g(x) = x^3 - 2x^2 + 6x - 12$. Our mission, should we choose to accept it, is to find the values of $x$ that make this function equal to zero. To help us on our quest, we'll be using a handy tool called the Rational Zeros Theorem. So, buckle up, and let's get started!

Understanding the Rational Zeros Theorem

So, what exactly is this Rational Zeros Theorem? Well, it's a neat little trick that helps us narrow down the possible rational zeros of a polynomial. Remember, rational numbers are those that can be expressed as a fraction (p/q), where p and q are integers. The theorem states that if a polynomial has integer coefficients, then any rational zero must be of the form p/q, where 'p' is a factor of the constant term (the term without any x) and 'q' is a factor of the leading coefficient (the coefficient of the term with the highest power of x).

Let's break this down for our specific polynomial, $g(x) = x^3 - 2x^2 + 6x - 12$. The constant term is -12, and the leading coefficient is 1 (since the coefficient of $x^3$ is 1). Now, we need to find the factors of both -12 and 1.

• Factors of -12: ±1, ±2, ±3, ±4, ±6, ±12
• Factors of 1: ±1

According to the Rational Zeros Theorem, any rational zero of our polynomial must be in the form p/q, where p is a factor of -12 and q is a factor of 1. This means our possible rational zeros are:

±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±12/1

Which simplifies to:

±1, ±2, ±3, ±4, ±6, ±12

That's quite a list, but it's much smaller than the infinite possibilities we started with! Now, we need to test these potential zeros to see which ones actually make our polynomial equal to zero. We can do this by plugging each value into the function and seeing if the result is zero. This process is called synthetic division or direct substitution.

Applying Synthetic Division to Find Zeros

Let's start by testing x = 2. We'll use synthetic division, a shortcut method for dividing a polynomial by a linear factor (x - c), where c is a potential zero.

Here's how synthetic division works:

1. Write down the coefficients of the polynomial: 1, -2, 6, -12
2. Write the potential zero (2) to the left.
3. Bring down the first coefficient (1).
4. Multiply the potential zero (2) by the number you just brought down (1), and write the result (2) under the next coefficient (-2).
5. Add the two numbers in the column (-2 and 2) and write the sum (0) below.
6. Repeat steps 4 and 5 until you've processed all the coefficients.

2 | 1  -2   6  -12
  |      2   0   12
  ----------------
    1   0   6    0

The last number in the bottom row is the remainder. If the remainder is 0, then the potential zero is indeed a zero of the polynomial. In this case, the remainder is 0, so x = 2 is a zero of $g(x)$. Awesome!

But wait, there's more! The other numbers in the bottom row (1, 0, 6) represent the coefficients of the quotient polynomial. Since we started with a cubic polynomial ($x^3$), the quotient polynomial is a quadratic ($x^2$). So, we have:

$x^2 + 0x + 6 = x^2 + 6$

This means we can rewrite our original polynomial as:

$g(x) = (x - 2)(x^2 + 6)$

Finding the Remaining Zeros

We've found one zero (x = 2), but we need to find all the zeros. To do this, we need to find the zeros of the quadratic factor, $x^2 + 6$. We can do this by setting it equal to zero and solving for x:

$x^2 + 6 = 0$ $x^2 = -6$ $x = ±√(-6)$ $x = ±√6 * √(-1)$ $x = ±√6i$

Here, we encounter imaginary numbers (i), since we're taking the square root of a negative number. So, the other two zeros are complex: x = √6i and x = -√6i. These are not rational zeros, which is why they didn't show up in our list from the Rational Zeros Theorem.

Final Answer

So, the zeros of the polynomial $g(x) = x^3 - 2x^2 + 6x - 12$ are:

• 2
• √6i
• -√6i

Therefore, the correct answer from the given choices is the one that lists these zeros. It's important to note that complex roots always come in conjugate pairs, which means if √6i is a root, then -√6i must also be a root.

In conclusion, the Rational Zeros Theorem is a powerful tool for finding potential rational zeros of a polynomial, but it doesn't give us all the zeros, especially if there are complex or irrational roots. For those, we might need to use other techniques like the quadratic formula or numerical methods. Keep practicing, and you'll become a zero-finding pro in no time!

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Understanding the Rational Zeros Theorem: In this section, we've thoroughly explained the Rational Zeros Theorem, emphasizing its importance in finding rational roots of polynomial equations. The theorem provides a systematic approach to narrow down the possible rational zeros, which significantly reduces the search space. The Rational Zeros Theorem states that if a polynomial has integer coefficients, then any rational zero must be of the form p/q, where 'p' is a factor of the constant term and 'q' is a factor of the leading coefficient. This theorem is crucial because it allows us to make an educated guess about the potential rational roots, rather than randomly trying different values. To illustrate this, consider our example polynomial $g(x) = x^3 - 2x^2 + 6x - 12$. The constant term is -12, and the leading coefficient is 1. By listing the factors of -12 (±1, ±2, ±3, ±4, ±6, ±12) and the factors of 1 (±1), we create a list of possible rational zeros. These possible zeros include ±1, ±2, ±3, ±4, ±6, and ±12. This process significantly streamlines our approach, as we only need to test these values to see if they are actual zeros of the polynomial. The Rational Zeros Theorem is a cornerstone in polynomial algebra, making the process of finding roots more manageable and efficient. This foundational step is indispensable before delving into methods like synthetic division or direct substitution. It is also worth noting that the Rational Zeros Theorem does not guarantee that we will find all the roots, as some roots may be irrational or complex. However, it provides a strong starting point for our analysis, especially when dealing with higher-degree polynomials where guessing is not a viable strategy. By understanding and applying this theorem, we can tackle a wide range of polynomial equations with increased confidence and accuracy. The theorem effectively bridges the gap between abstract polynomial equations and concrete numerical solutions, making it an essential tool in every mathematician's toolkit. Moreover, the Rational Zeros Theorem often serves as a stepping stone to more advanced techniques in algebra, such as numerical approximations for irrational roots or graphical methods for visualizing polynomial behavior. It is a testament to the power of mathematical theory in simplifying complex problems and providing a structured approach to problem-solving. In essence, the Rational Zeros Theorem is not just a theorem; it is a strategy, a method, and a foundation for further exploration in the realm of polynomials.

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