Polynomial Factor: Find It Now!
Hey guys! Let's dive into the exciting world of polynomials and figure out how to find their factors. Today, we're tackling a specific problem: identifying a factor of the polynomial f(x) = 6x⁴ - 21x³ - 4x² + 24x - 35. This might seem daunting at first, but don't worry, we'll break it down step by step. Understanding polynomial factorization is crucial in various areas of mathematics, from solving equations to simplifying expressions. So, grab your thinking caps, and let's get started!
Understanding Polynomial Factors
Before we jump into solving the problem, let's make sure we're all on the same page about what a polynomial factor actually is. Essentially, a factor of a polynomial is another polynomial that divides evenly into the original polynomial, leaving no remainder. Think of it like finding the factors of a regular number, but with algebraic expressions. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because 12 can be divided evenly by each of these numbers. Similarly, if (x - a) is a factor of f(x), then f(a) will equal zero. This is a key concept we'll use to solve our problem.
When dealing with polynomials, several techniques can help us identify factors. These include:
- Factoring by Grouping: This involves grouping terms together and factoring out common factors. It's useful when you can easily spot common factors within different parts of the polynomial.
- The Rational Root Theorem: This theorem helps us identify potential rational roots (and thus factors) of the polynomial. It's a more systematic way to narrow down our search.
- Synthetic Division: A quick and efficient way to test potential roots and divide the polynomial by a linear factor.
- The Factor Theorem: This is the core principle we mentioned earlier: if f(a) = 0, then (x - a) is a factor of f(x). We'll be using this extensively.
Understanding these methods will make the process of finding polynomial factors much smoother. Now, let's apply these ideas to our specific problem.
The Problem: f(x) = 6x⁴ - 21x³ - 4x² + 24x - 35
Our mission, should we choose to accept it, is to find a factor of the polynomial f(x) = 6x⁴ - 21x³ - 4x² + 24x - 35. We're given four possible factors to choose from:
- A. 2x - 7
- B. 2x + 7
- C. 3x - 7
- D. 3x + 7
To solve this, we'll primarily use the Factor Theorem. Remember, if (ax - b) is a factor of f(x), then f(b/a) must equal zero. This means we need to test each option by setting the potential factor equal to zero, solving for x, and then plugging that x value into our polynomial.
This might sound like a bit of work, but it's a straightforward process. We'll take each option one by one and see if it fits the bill. Let's get to it!
Testing the Options
Let's systematically test each of the given options to see which one is a factor of our polynomial. We'll use the Factor Theorem, which, as we discussed, states that if f(b/a) = 0 for a factor (ax - b), then (ax - b) is indeed a factor of f(x).
Option A: 2x - 7
- Set the potential factor equal to zero: 2x - 7 = 0
- Solve for x: 2x = 7 => x = 7/2
- Substitute x = 7/2 into f(x): f(7/2) = 6(7/2)⁴ - 21(7/2)³ - 4(7/2)² + 24(7/2) - 35
Let's calculate this step by step:
f(7/2) = 6(2401/16) - 21(343/8) - 4(49/4) + 24(7/2) - 35 f(7/2) = (6 * 2401) / 16 - (21 * 343) / 8 - (4 * 49) / 4 + (24 * 7) / 2 - 35 f(7/2) = 14406 / 16 - 7203 / 8 - 196 / 4 + 168 / 2 - 35
To make the arithmetic easier, let’s convert all fractions to have a common denominator of 16:
f(7/2) = 14406 / 16 - (7203 * 2) / 16 - (196 * 4) / 16 + (168 * 8) / 16 - (35 * 16) / 16 f(7/2) = 14406 / 16 - 14406 / 16 - 784 / 16 + 1344 / 16 - 560 / 16 f(7/2) = (14406 - 14406 - 784 + 1344 - 560) / 16 f(7/2) = (0) / 16 f(7/2) = 0
Since f(7/2) = 0, 2x - 7 is a factor of the polynomial. Hooray! We found our answer, but let's quickly check the other options just to be thorough.
Option B: 2x + 7
- Set the potential factor equal to zero: 2x + 7 = 0
- Solve for x: 2x = -7 => x = -7/2
- Substitute x = -7/2 into f(x): f(-7/2) = 6(-7/2)⁴ - 21(-7/2)³ - 4(-7/2)² + 24(-7/2) - 35
Without going through the full calculation (which you can do yourself to practice!), you'll find that f(-7/2) is not equal to zero. Therefore, 2x + 7 is not a factor.
Option C: 3x - 7
- Set the potential factor equal to zero: 3x - 7 = 0
- Solve for x: 3x = 7 => x = 7/3
- Substitute x = 7/3 into f(x): f(7/3) = 6(7/3)⁴ - 21(7/3)³ - 4(7/3)² + 24(7/3) - 35
Again, without the full calculation, f(7/3) will not equal zero, so 3x - 7 is not a factor.
Option D: 3x + 7
- Set the potential factor equal to zero: 3x + 7 = 0
- Solve for x: 3x = -7 => x = -7/3
- Substitute x = -7/3 into f(x): f(-7/3) = 6(-7/3)⁴ - 21(-7/3)³ - 4(-7/3)² + 24(-7/3) - 35
Similarly, f(-7/3) will not equal zero, meaning 3x + 7 is not a factor.
The Solution: Option A is the Winner!
Through our systematic testing, we've confirmed that Option A, 2x - 7, is indeed a factor of the polynomial f(x) = 6x⁴ - 21x³ - 4x² + 24x - 35. We achieved this by using the Factor Theorem and diligently evaluating the polynomial at the appropriate x-values.
Key Takeaways and Further Practice
- The Factor Theorem is your best friend when finding polynomial factors. Remember, if f(a) = 0, then (x - a) is a factor.
- Systematic testing is crucial. Don't just guess! Work through each option carefully.
- Arithmetic matters! Make sure your calculations are accurate to avoid errors.
To further sharpen your skills, try applying these techniques to other polynomial factorization problems. You can find plenty of examples online or in textbooks. The more you practice, the more comfortable you'll become with these concepts. You've got this!
Polynomial factorization might seem intimidating at first, but with the right tools and a bit of practice, it becomes much more manageable. Keep exploring, keep learning, and never stop challenging yourself. Happy factoring!