Solving for k in Polynomial Division: A Problem in Algebra

Polynomial division is an essential skill that plays a crucial role in advanced mathematics. One common question in algebra involves finding the value of a constant when the division of a polynomial by a linear expression yields a particular remainder. In this article, we will explore a problem of this nature and apply the remainder theorem to find the unknown constant. By the end of this discussion, you will be able to master the steps required to solve similar problems.

Problem Statement

The given problem is: When 2x3kx22 is divided by x - 1, the remainder is 7. What must be the value of k?

Understanding the Remainder Theorem

The remainder theorem states that if a polynomial f(x) is divided by a linear expression x - a, the remainder of this division is f(a). This theorem is particularly useful in solving polynomial division problems.

Solving the Problem Step-by-Step

Step 1: Write the Polynomial Expression
The given polynomial expression is 2x3kx22. Let's denote this polynomial as f(x) 2x3kx22.

Step 2: Apply the Remainder Theorem
According to the remainder theorem, we need to find the value of the polynomial when x 1. This is because we are dividing by x - 1.

Substitute x 1 into the polynomial:

f(1) 2(1)3k(1)22

Step 3: Simplify the Expression
Simplify the expression using the properties of exponents. Since any number raised to the power of 1 is the number itself, we have:

2(1)3(1)22 2(1) 2

Step 4: Equate to the Given Remainder
The problem states that the remainder is 7. Therefore, we can set the simplified expression equal to 7:

2 7

This is clearly not correct. We must have made a mistake in understanding the problem. Let's re-examine the given polynomial. The polynomial should be 2x3kx22, which simplifies to:

2(1)3(1)k22 2k2

Step 5: Solve for k
Now, we can set the simplified expression equal to 7:

2k2 7

Solving for k, we get:

k2 3.5

k √(3.5) ≈ 1.87

However, the problem likely intended for the polynomial to be written as 2x3x22, which simplifies to:

2x25

Substituting x 1 into this simplified polynomial:

f(1) 2(1)25 2

To get the remainder as 7, we need:

2k2 7

Solving for k, we get:

k2 3.5
k √(3.5) ≈ 1.87

For simplicity and to match the given solution, we often round or simplify the constants. However, in this case, the exact solution involves a square root.

Conclusion

By applying the remainder theorem and simplifying the polynomial expression, we can solve for the unknown constant k. This problem demonstrates the importance of careful algebraic manipulation and the application of fundamental theorems in solving polynomial division problems.

Additional Insights

1. **Polynomial Division Basics**: Understanding the basics of polynomial division, including the division algorithm and long division, is crucial for solving such problems.

2. **Remainder Theorem**: Familiarity with the remainder theorem simplifies the process of finding remainders in polynomial division without performing the full division.

3. **Algebraic Manipulation**: Mastering algebraic manipulation, especially with exponents and constants, is key to solving complex algebraic problems.

Keywords: polynomial division, algebra, remainder theorem