Finding Zeros Of Polynomial Q(x) = 3x⁴ + 13x³ - 3x² - 33x - 20
In the realm of mathematics, particularly in algebra, finding the zeros of a polynomial function is a fundamental task. The zeros, also known as roots or x-intercepts, are the values of x that make the polynomial equal to zero. This process is crucial for understanding the behavior of polynomial functions and their graphs, as well as for solving various mathematical problems. In this article, we will delve into the step-by-step process of finding all the zeros of the polynomial function q(x) = 3x⁴ + 13x³ - 3x² - 33x - 20, expressing the answers in exact form. This exploration will involve utilizing the Rational Root Theorem, synthetic division, and factoring techniques to arrive at the solutions. Understanding these methods is essential for anyone studying algebra and seeking to master polynomial equations.
Understanding the Problem: q(x) = 3x⁴ + 13x³ - 3x² - 33x - 20
Before we begin to solve, let's clearly understand the polynomial function we are dealing with. We have q(x) = 3x⁴ + 13x³ - 3x² - 33x - 20. This is a quartic polynomial, meaning the highest power of x is 4. Consequently, by the Fundamental Theorem of Algebra, we expect to find up to four complex zeros (which may include real zeros). The coefficients of the polynomial are integers, which allows us to use the Rational Root Theorem to narrow down the possible rational zeros. Identifying the polynomial and its characteristics is the initial step in our journey to finding the zeros. The Rational Root Theorem will be our guiding principle, helping us systematically test potential solutions and reduce the complexity of the equation. Our ultimate goal is to express these zeros in their exact form, which means avoiding decimal approximations and retaining any radical or fractional components.
Step 1: Applying the Rational Root Theorem
The Rational Root Theorem is a powerful tool for identifying potential rational zeros of a polynomial. It states that if a polynomial with integer coefficients has rational roots, they 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. In our case, the constant term is -20 and the leading coefficient is 3. Let's list the factors of each:
- Factors of -20 (p): ±1, ±2, ±4, ±5, ±10, ±20
- Factors of 3 (q): ±1, ±3
Now, we form all possible fractions p/q: ±1, ±2, ±4, ±5, ±10, ±20, ±1/3, ±2/3, ±4/3, ±5/3, ±10/3, ±20/3. This gives us a list of potential rational zeros that we can test. While it seems like a long list, it's significantly smaller than the infinite possibilities if we didn't have this theorem. This step is crucial because it provides a systematic way to approach the problem, rather than randomly guessing solutions. The Rational Root Theorem acts as a filter, narrowing our search and making the process of finding zeros more efficient.
Step 2: Testing Potential Zeros with Synthetic Division
With our list of potential rational zeros from the Rational Root Theorem, we now need to test them. Synthetic division is an efficient method for this. Let's start by testing x = -1: Using synthetic division with -1, we find:
-1 | 3 13 -3 -33 -20
| -3 -10 13 20
------------------------
3 10 -13 -20 0
The remainder is 0, which means x = -1 is a zero, and (x + 1) is a factor of q(x). The quotient, 3x³ + 10x² - 13x - 20, represents the remaining polynomial after dividing out the factor (x + 1). This is a significant step because it reduces the degree of the polynomial, making it easier to find the remaining zeros. We can now focus on the cubic polynomial, which is simpler to work with than the original quartic. The synthetic division not only confirms whether a potential zero is an actual zero but also provides the coefficients of the resulting polynomial, streamlining the process of finding further zeros. This iterative approach is key to solving higher-degree polynomial equations.
Step 3: Continuing Synthetic Division or Factoring
We now have the cubic polynomial 3x³ + 10x² - 13x - 20. We can continue using synthetic division to test the remaining potential rational zeros from our list. Let's try x = -4/3:
-4/3 | 3 10 -13 -20
| -4 -8 28
------------------
3 6 -21 8
The remainder is 0, so x = -4/3 is not a zero. So, let's try x = -5/3:
-5/3 | 3 10 -13 -20
| -5 -25/3 95/9
------------------
3 5 -64/3 5/9
The remainder is not 0, so x = -5/3 is not a zero. Let's test x = -5:
-5 | 3 10 -13 -20
| -15 25 -60
------------------
3 -5 12 -80
The remainder is not 0, so x = -5 is not a zero. Let's test x = 4/3:
-4/3 | 3 10 -13 -20
| -4 -8 28
------------------
3 6 -21 8
The remainder is not 0, so x = 4/3 is not a zero. So, let's try x = 5/3:
-5/3 | 3 10 -13 -20
| -5 -25/3 95/9
------------------
3 5 -64/3 5/9
The remainder is not 0, so x = 5/3 is not a zero. So, let's try x = -4:
-4 | 3 10 -13 -20
| -12 8 20
------------------
3 -2 -5 0
The remainder is 0, which means x = -4 is a zero, and (x + 4) is a factor. Now we have a quadratic quotient: 3x² - 2x - 5. This quadratic can be factored further or solved using the quadratic formula. The decision to continue with synthetic division or switch to factoring often depends on the complexity of the resulting polynomial. In this case, we've reduced the cubic to a quadratic, which is much more manageable. The ability to identify and apply the most efficient method at each stage is a key skill in solving polynomial equations.
Step 4: Solving the Quadratic Equation
We are left with the quadratic equation 3x² - 2x - 5 = 0. We can try to factor it:
(3x - 5)(x + 1) = 0
Setting each factor to zero gives us:
- 3x - 5 = 0 => x = 5/3
- x + 1 = 0 => x = -1
So, the zeros from the quadratic factor are x = 5/3 and x = -1. This step demonstrates the importance of factoring skills in solving polynomial equations. When a polynomial can be factored, it simplifies the process of finding zeros considerably. However, if factoring is not straightforward, the quadratic formula provides a reliable alternative. Mastering both factoring and the quadratic formula is essential for handling quadratic equations effectively.
Step 5: Listing All Zeros
Combining all the zeros we found:
- From synthetic division: x = -1 and x = -4
- From factoring the quadratic: x = 5/3 and x = -1
We have a repeated zero of x = -1. Therefore, the zeros of q(x) = 3x⁴ + 13x³ - 3x² - 33x - 20 are x = -4, -1, 5/3. Note that x=-1 has a multiplicity of 2.
The importance of listing all zeros cannot be overstated. We must account for any repeated roots, as they affect the behavior of the polynomial graph at those points. Additionally, ensuring that we have found the expected number of roots (based on the degree of the polynomial) is a crucial verification step. In this case, we have a quartic polynomial, so we expect four roots, counting multiplicities. Our solution satisfies this criterion, giving us confidence in the accuracy of our result.
Conclusion
In summary, we found all the zeros of the polynomial function q(x) = 3x⁴ + 13x³ - 3x² - 33x - 20 to be x = -4, -1, 5/3, with x = -1 having a multiplicity of 2. This process involved using the Rational Root Theorem to identify potential rational zeros, synthetic division to test these potentials and reduce the polynomial's degree, and factoring techniques to solve the resulting quadratic equation. This comprehensive approach demonstrates the power of combining different algebraic methods to tackle polynomial equations. The journey of finding the zeros of a polynomial function is not just about arriving at the correct answer; it's about developing a deep understanding of the underlying principles and techniques that can be applied to a wide range of mathematical problems. By mastering these skills, students can confidently navigate the complexities of algebra and beyond.
