Determining the Value of K for Simultaneous x-2 as a Factor in 3x^4 - 8x^2 - kx^6

Are you working on a challenging polynomial equation where you need to find the value of k that makes x-2 a factor? This article will guide you through the steps to determine this value accurately. Let's dive in!

Understanding the Problem

The problem at hand is to find the value of k so that x-2 is a factor of the polynomial 3x^4 - 8x^2 - kx^6. This means that if you substitute x2 into the equation, it should equal zero.

Substituting x 2

Let's start by substituting x 2 into the polynomial and setting it equal to zero to solve for k.

Given the polynomial:

[f(x) 3x^4 - 8x^2 - kx^6]

Substitute x 2:

[f(2) 3(2)^4 - 8(2)^2 - k(2)^6]

This simplifies to:

[f(2) 3(16) - 8(4) - k(64)]

Further simplifying:

[f(2) 48 - 32 - 64k]

Setting f(2) equal to zero:

[48 - 32 - 64k 0]

Solving for k:

[-64k -16]

[k frac{-16}{-64}]

[k frac{1}{4}]

Therefore, the value of k is frac{1}{4}.

Conclusion

In conclusion, the value of k that makes x-2 a factor of the polynomial 3x^4 - 8x^2 - kx^6 is frac{1}{4}. This ensures that when x2, the polynomial equals zero, confirming that x-2 is indeed a factor.

If you have any further questions or need more assistance with similar problems, feel free to reach out!