Solving for K in a Polynomial with Given Factor

In the realm of polynomial algebra, identifying unknown coefficients is a common challenge. This article addresses the specific case of finding the value of K in the polynomial equation p(x) kx3 - 3x2 - 2kx 1, with a known factor of x 1. We will walk through the process step-by-step to determine the value of K using algebraic methods.

Step 1: Given Polynomial and Factor

The polynomial is given by:

p(x) kx3 - 3x2 - 2kx 1

We know that x 1 is a factor of this polynomial. Therefore, p(-1) 0.

Step 2: Substitute x -1 into the Polynomial

Substituting x -1 into the equation yields:

p(-1) k(-1)3 - 3(-1)2 - 2k(-1) 1

Simplifying the expression:

k(-1) - 3(1) 2k 1

-k - 3 2k 1

k - 2

Since p(-1) 0, we have:

k - 2 0

Solving for K:

k 2

Step 3: Verification and Generalization

To ensure the correctness of our solution, we can substitute K back into the original polynomial and check that x 1 is indeed a factor. The polynomial now is:

p(x) 2x3 - 3x2 - 4x 1

Again, substituting x -1 into this new polynomial:

p(-1) 2(-1)3 - 3(-1)2 - 4(-1) 1

2(-1) - 3(1) 4 1

-2 - 3 4 1

This confirms that x 1 is a factor and the value of K is indeed 2.

Conclusion

By substituting the known factor into the polynomial and solving the resulting equation, we have determined that the value of K is 2. This method can be applied to similar problems in polynomial algebra, providing a systematic approach to solving for unknown coefficients.

Additional Resources

For further exploration of polynomial factorization, factor theorem, and algebraic equation solving, you may refer to:

MathIsFun - Polynomials Khan Academy - Fundamental Theorem of Algebra Varsity Tutors - Solving Polynomial Equations by Factoring