Identifying Impossible Zeros Of Polynomials An In-Depth Look At F(x) = 4x³ - 5x² + 12
In the realm of polynomial functions, identifying zeros, also known as roots or x-intercepts, is a fundamental task. These zeros are the values of x for which the function f(x) equals zero. Knowing the zeros of a polynomial allows us to fully understand the behavior of the function and to solve various mathematical problems.
In this article, we'll dive deep into a specific problem: determining which of the given values (5/3, 2, 5/2, and 7) is not a possible zero of the cubic function f(x) = 4x³ - 5x² + 12. We'll explore the Rational Root Theorem, synthetic division, and direct substitution as powerful tools to tackle this challenge. By the end of this journey, you'll not only know the answer but also gain a solid understanding of the techniques involved in finding and verifying polynomial zeros.
Understanding the Question: Possible Zeros and the Rational Root Theorem
Before we jump into the calculations, let's clarify what the question is asking. We're looking for the value among the choices that cannot be a zero of the function f(x) = 4x³ - 5x² + 12. This means that when we substitute that value for x in the function, the result will not be zero.
To approach this problem systematically, we can use the Rational Root Theorem. This theorem provides a list of potential rational zeros of a polynomial function with integer coefficients. It states that if a rational number p/q (in lowest terms) is a root of the polynomial:
aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀,
then p must be a factor of the constant term a₀, and q must be a factor of the leading coefficient aₙ.
In our case, f(x) = 4x³ - 5x² + 12, the constant term a₀ is 12, and the leading coefficient aₙ is 4. Therefore, the possible rational roots are of the form p/q, where p is a factor of 12 (±1, ±2, ±3, ±4, ±6, ±12) and q is a factor of 4 (±1, ±2, ±4). Listing out all possible combinations, we get:
±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2, ±1/4, ±3/4
This list gives us a starting point for checking the given options. Notice that 5/3, 2, and 5/2 are all on the list of possible rational roots, but 7 is not. This might lead us to suspect that 7 is the answer, but we still need to verify. The Rational Root Theorem only gives us a list of possible rational roots; it doesn't guarantee that any of them are actual roots.
Method 1: Direct Substitution – A Straightforward Approach
The most direct way to check if a value is a zero of a function is to simply substitute the value into the function and see if the result is zero. Let's apply this method to each of the given options.
Option A: x = 5/3
Substitute x = 5/3 into f(x) = 4x³ - 5x² + 12:
f(5/3) = 4(5/3)³ - 5(5/3)² + 12
f(5/3) = 4(125/27) - 5(25/9) + 12
f(5/3) = 500/27 - 125/9 + 12
To combine these terms, we need a common denominator, which is 27:
f(5/3) = 500/27 - 375/27 + 324/27
f(5/3) = 449/27
Since f(5/3) is not equal to zero, 5/3 is not a zero of the function. However, we are looking for the value that is not a possible zero, and we haven't proven that 5/3 is impossible yet, just that it's not a root.
Option B: x = 2
Substitute x = 2 into f(x) = 4x³ - 5x² + 12:
f(2) = 4(2)³ - 5(2)² + 12
f(2) = 4(8) - 5(4) + 12
f(2) = 32 - 20 + 12
f(2) = 24
Again, f(2) is not equal to zero, so 2 is not a zero of the function. But like 5/3, it's not impossible based on the Rational Root Theorem.
Option C: x = 5/2
Substitute x = 5/2 into f(x) = 4x³ - 5x² + 12:
f(5/2) = 4(5/2)³ - 5(5/2)² + 12
f(5/2) = 4(125/8) - 5(25/4) + 12
f(5/2) = 125/2 - 125/4 + 12
To combine these terms, we need a common denominator, which is 4:
f(5/2) = 250/4 - 125/4 + 48/4
f(5/2) = 173/4
Since f(5/2) is not equal to zero, 5/2 is not a zero of the function either. Again, it's a possible zero according to the Rational Root Theorem.
Option D: x = 7
Substitute x = 7 into f(x) = 4x³ - 5x² + 12:
f(7) = 4(7)³ - 5(7)² + 12
f(7) = 4(343) - 5(49) + 12
f(7) = 1372 - 245 + 12
f(7) = 1139
Since f(7) is not equal to zero, 7 is not a zero of the function. And, crucially, 7 was not on our list of possible rational roots generated by the Rational Root Theorem.
Based on our direct substitution, none of the values are zeros of the function. However, only 7 is not a possible rational root according to the Rational Root Theorem. This makes 7 the most likely answer.
Method 2: Synthetic Division – A More Efficient Approach
While direct substitution works, it can be computationally intensive, especially for higher-degree polynomials. Synthetic division is a more efficient method for testing potential roots. It's a streamlined way to divide a polynomial by a linear factor of the form (x - c), where 'c' is the potential root.
If the remainder after synthetic division is zero, then 'c' is a root of the polynomial. If the remainder is not zero, then 'c' is not a root.
Let's apply synthetic division to our problem, testing each of the given options. We'll set up the synthetic division table using the coefficients of our polynomial f(x) = 4x³ - 5x² + 0x + 12 (note the inclusion of the 0x term as a placeholder):
| 4 | -5 | 0 | 12 | |
|---|---|---|---|---|
| c | ||||
| Rem |
Option A: x = 5/3
| 4 | -5 | 0 | 12 | |
|---|---|---|---|---|
| 5/3 | 20/3 | 175/9 | 875/27 | |
| 4 | 5/3 | 175/9 | 1199/27 |
The remainder is 1199/27, which is not zero. Therefore, 5/3 is not a zero.
Option B: x = 2
| 4 | -5 | 0 | 12 | |
|---|---|---|---|---|
| 2 | 8 | 6 | 12 | |
| 4 | 3 | 6 | 24 |
The remainder is 24, which is not zero. Therefore, 2 is not a zero.
Option C: x = 5/2
| 4 | -5 | 0 | 12 | |
|---|---|---|---|---|
| 5/2 | 10 | 25/2 | 625/4 | |
| 4 | 5 | 25/2 | 673/4 |
The remainder is 673/4, which is not zero. Therefore, 5/2 is not a zero.
Option D: x = 7
| 4 | -5 | 0 | 12 | |
|---|---|---|---|---|
| 7 | 28 | 161 | 1127 | |
| 4 | 23 | 161 | 1139 |
The remainder is 1139, which is not zero. Therefore, 7 is not a zero.
Again, synthetic division confirms that none of the values are zeros of the function. However, it reinforces the fact that 7 is not a possible rational root, making it the answer we're looking for.
Conclusion: Identifying the Impossible Zero
Through both direct substitution and synthetic division, we've confirmed that none of the given values are actual zeros of the function f(x) = 4x³ - 5x² + 12. However, by applying the Rational Root Theorem, we identified that 7 is the only value that is not a possible rational zero. Therefore, the answer to the question, "Which of the following is not a possible zero of f(x) = 4x³ - 5x² + 12?" is:
D. 7
This problem demonstrates the power of the Rational Root Theorem in narrowing down the potential zeros of a polynomial. While it doesn't tell us the actual roots, it provides a valuable list to test. When combined with methods like direct substitution or synthetic division, we can efficiently determine the zeros of a polynomial and identify values that simply cannot be roots.
Understanding these techniques is crucial for solving a wide range of problems in algebra and calculus. By mastering these concepts, you'll be well-equipped to tackle more complex polynomial equations and gain a deeper understanding of the behavior of functions.
