Factor Theorem: Is 2x-7 A Factor? Polynomial Explained

by Chloe Fitzgerald 55 views

Hey guys! Today, we're diving into the fascinating world of polynomials, specifically tackling the question: Which of these options is a factor of the polynomial f(x) = 6x⁴ - 21x³ - 4x² + 24x - 35? It might look intimidating at first, but don't worry, we'll break it down step by step.

Understanding the Factor Theorem

Before we jump into the solution, let's quickly recap the Factor Theorem. This theorem is our trusty tool in this situation. It states that for a polynomial f(x), if f(a) = 0 for some value a, then (x - a) is a factor of f(x). Conversely, if (x - a) is a factor of f(x), then f(a) = 0. Basically, we're looking for a value of x that makes the polynomial equal to zero. This value will then help us identify the correct factor.

Applying the Factor Theorem to Our Problem

Now, let's apply this to our polynomial, f(x) = 6x⁴ - 21x³ - 4x² + 24x - 35. We're given four possible factors:

  • A. 2x - 7
  • B. 2x + 7
  • C. 3x - 7
  • D. 3x + 7

To use the Factor Theorem, we need to find the values of x that would make each of these factors equal to zero. Let's do that for each option:

  • For 2x - 7 = 0, we get x = 7/2
  • For 2x + 7 = 0, we get x = -7/2
  • For 3x - 7 = 0, we get x = 7/3
  • For 3x + 7 = 0, we get x = -7/3

So, we have four potential values of x: 7/2, -7/2, 7/3, and -7/3. The next step is to plug each of these values into our polynomial f(x) and see which one makes it equal to zero. This might seem a bit tedious, but it's the most straightforward way to solve this problem.

Testing the Potential Factors

Let's start with option A, where x = 7/2. We need to calculate f(7/2):

f(7/2) = 6(7/2)⁴ - 21(7/2)³ - 4(7/2)² + 24(7/2) - 35

This looks like a lot of arithmetic, but let's break it down:

  • (7/2)⁴ = 2401/16
  • (7/2)³ = 343/8
  • (7/2)² = 49/4

Plugging these back into the equation, we get:

f(7/2) = 6(2401/16) - 21(343/8) - 4(49/4) + 24(7/2) - 35

Simplifying further:

f(7/2) = 14406/16 - 7203/8 - 49 + 84 - 35

To make the fractions easier to deal with, let's convert everything to have a denominator of 16:

f(7/2) = 14406/16 - 14406/16 - 784/16 + 1344/16 - 560/16

Now we can combine the terms:

f(7/2) = (14406 - 14406 - 784 + 1344 - 560) / 16

f(7/2) = 0 / 16 = 0

Bingo! We found that f(7/2) = 0. According to the Factor Theorem, this means that (x - 7/2) is a factor of f(x). But wait, our options are in the form of ax + b, so we need to do a little more work.

Connecting the Dots

Since (x - 7/2) is a factor, we can multiply the whole expression by 2 to get rid of the fraction, resulting in (2x - 7). So, 2x - 7 is indeed a factor of our polynomial. We've found our answer, but let's quickly discuss why the other options are incorrect and reinforce our understanding of the process.

Why the Other Options Don't Work

Although we've already identified the correct answer, it's beneficial to understand why the other options don't fit. This solidifies our comprehension of the Factor Theorem and the process involved.

Option B: 2x + 7

For the factor (2x + 7), we need to test x = -7/2. Let's calculate f(-7/2):

f(-7/2) = 6(-7/2)⁴ - 21(-7/2)³ - 4(-7/2)² + 24(-7/2) - 35

This is where paying attention to signs becomes crucial. Recall that a negative number raised to an even power is positive, and a negative number raised to an odd power is negative. Breaking down the terms:

  • (-7/2)⁴ = 2401/16
  • (-7/2)³ = -343/8
  • (-7/2)² = 49/4

Plugging these back into the equation:

f(-7/2) = 6(2401/16) - 21(-343/8) - 4(49/4) + 24(-7/2) - 35

Simplifying:

f(-7/2) = 14406/16 + 7203/8 - 49 - 84 - 35

Converting to a common denominator of 16:

f(-7/2) = 14406/16 + 14406/16 - 784/16 - 1344/16 - 560/16

Combining the terms:

f(-7/2) = (14406 + 14406 - 784 - 1344 - 560) / 16

f(-7/2) = 26124 / 16 ≠ 0

Since f(-7/2) is not equal to zero, (2x + 7) is not a factor of the polynomial.

Option C: 3x - 7

For the factor (3x - 7), we need to test x = 7/3. Let's calculate f(7/3):

f(7/3) = 6(7/3)⁴ - 21(7/3)³ - 4(7/3)² + 24(7/3) - 35

Breaking down the terms:

  • (7/3)⁴ = 2401/81
  • (7/3)³ = 343/27
  • (7/3)² = 49/9

Plugging these back into the equation:

f(7/3) = 6(2401/81) - 21(343/27) - 4(49/9) + 24(7/3) - 35

Simplifying:

f(7/3) = 14406/81 - 7203/27 - 196/9 + 56 - 35

Converting to a common denominator of 81:

f(7/3) = 14406/81 - 21609/81 - 1764/81 + 4536/81 - 2835/81

Combining the terms:

f(7/3) = (14406 - 21609 - 1764 + 4536 - 2835) / 81

f(7/3) = -7266 / 81 ≠ 0

Since f(7/3) is not equal to zero, (3x - 7) is not a factor of the polynomial.

Option D: 3x + 7

For the factor (3x + 7), we need to test x = -7/3. Let's calculate f(-7/3):

f(-7/3) = 6(-7/3)⁴ - 21(-7/3)³ - 4(-7/3)² + 24(-7/3) - 35

Breaking down the terms:

  • (-7/3)⁴ = 2401/81
  • (-7/3)³ = -343/27
  • (-7/3)² = 49/9

Plugging these back into the equation:

f(-7/3) = 6(2401/81) - 21(-343/27) - 4(49/9) + 24(-7/3) - 35

Simplifying:

f(-7/3) = 14406/81 + 7203/27 - 196/9 - 56 - 35

Converting to a common denominator of 81:

f(-7/3) = 14406/81 + 21609/81 - 1764/81 - 4536/81 - 2835/81

Combining the terms:

f(-7/3) = (14406 + 21609 - 1764 - 4536 - 2835) / 81

f(-7/3) = 26880 / 81 ≠ 0

Since f(-7/3) is not equal to zero, (3x + 7) is not a factor of the polynomial.

Conclusion

Therefore, the correct answer is A. 2x - 7. We successfully used the Factor Theorem to determine which of the given options is a factor of the polynomial f(x) = 6x⁴ - 21x³ - 4x² + 24x - 35. By plugging in the values of x derived from each potential factor, we found that f(7/2) = 0, confirming that (2x - 7) is indeed a factor. This exercise highlights the power and utility of the Factor Theorem in polynomial factorization. Keep practicing, guys, and you'll become polynomial pros in no time!