Finding The Value Of K In The Polynomial 9x³ - Kx + 4

In the realm of algebra, polynomials form the bedrock of many mathematical concepts and applications. Understanding the relationship between factors and roots of polynomials is crucial for solving a wide range of problems. This article delves into a specific problem involving a cubic polynomial and one of its factors. We will explore how to determine the value of an unknown coefficient within the polynomial, given that a linear expression is a factor.

Problem Statement

We are given the polynomial 9x³ - kx + 4, where k is an integer. We also know that 3x - 2 is a factor of this polynomial. Our objective is to find the value of k.

Understanding the Factor Theorem

The Factor Theorem is a fundamental concept that links the factors of a polynomial to its roots. It states that for a polynomial p(x), if (x - a) is a factor, then p(a) = 0. Conversely, if p(a) = 0, then (x - a) is a factor of p(x). This theorem provides a powerful tool for solving problems involving polynomial factorization.

In our case, since 3x - 2 is a factor of the polynomial 9x³ - kx + 4, we can use the Factor Theorem to find the value of x that makes the factor equal to zero. Setting 3x - 2 = 0, we solve for x:

3x - 2 = 0 3x = 2 x = 2/3

Thus, x = 2/3 is a root of the polynomial 9x³ - kx + 4. This means that if we substitute x = 2/3 into the polynomial, the result should be zero.

Applying the Factor Theorem to Find k

Now, we substitute x = 2/3 into the polynomial 9x³ - kx + 4:

9(2/3)³ - k(2/3) + 4 = 0

Let's simplify this equation step by step:

9(8/27) - (2/3)k + 4 = 0 (8/3) - (2/3)k + 4 = 0

To eliminate the fractions, we can multiply the entire equation by 3:

3 * [(8/3) - (2/3)k + 4] = 3 * 0 8 - 2k + 12 = 0

Combine the constant terms:

20 - 2k = 0

Now, isolate the term with k:

2k = 20

Finally, solve for k:

k = 20 / 2 k = 10

Therefore, the value of k that makes 3x - 2 a factor of the polynomial 9x³ - kx + 4 is 10.

Verification

To verify our result, we can substitute k = 10 back into the original polynomial and perform polynomial division. The polynomial becomes 9x³ - 10x + 4. We will divide this polynomial by the factor 3x - 2.

Polynomial division can be performed using various methods, such as long division or synthetic division. Here, we will demonstrate the long division method.

 3x² + 2x - 2
3x - 2 | 9x³ + 0x² - 10x + 4
        -(9x³ - 6x²)
        ------------------
              6x² - 10x
              -(6x² - 4x)
              ------------------
                    -6x + 4
                    -(-6x + 4)
                    ------------------
                          0

As we can see, the remainder is 0, which confirms that 3x - 2 is indeed a factor of 9x³ - 10x + 4. The quotient is 3x² + 2x - 2. This verification step provides confidence in our solution for k.

Alternative Method: Synthetic Division

Synthetic division provides a more streamlined approach to dividing polynomials, particularly when the divisor is a linear expression. To use synthetic division, we first identify the root of the divisor, which we already found to be x = 2/3. Then, we set up the synthetic division table using the coefficients of the polynomial 9x³ - 10x + 4.

The coefficients are 9 (for x³), 0 (for x²), -10 (for x), and 4 (the constant term). Note the importance of including the 0 coefficient for the missing x² term.

2/3 | 9  0  -10  4
    |    6   4  -4
    ----------------
      9  6  -6   0

Here's how the synthetic division works:

1. Bring down the leading coefficient (9).
2. Multiply the root (2/3) by the brought-down coefficient (9) and write the result (6) under the next coefficient (0).
3. Add the numbers in the second column (0 + 6 = 6).
4. Multiply the root (2/3) by the result (6) and write the result (4) under the next coefficient (-10).
5. Add the numbers in the third column (-10 + 4 = -6).
6. Multiply the root (2/3) by the result (-6) and write the result (-4) under the last coefficient (4).
7. Add the numbers in the last column (4 + (-4) = 0).

The last number in the bottom row (0) is the remainder. The other numbers (9, 6, -6) are the coefficients of the quotient, which is 9x² + 6x - 6. Since the remainder is 0, this confirms that 3x - 2 is a factor, and our value of k = 10 is correct.

Conclusion

In this article, we successfully determined the value of k in the polynomial 9x³ - kx + 4, given that 3x - 2 is a factor. We utilized the Factor Theorem, which states that if (x - a) is a factor of a polynomial p(x), then p(a) = 0. By finding the root of the factor (3x - 2) and substituting it into the polynomial, we formed an equation and solved for k. We found that k = 10. Furthermore, we verified our result using polynomial long division and synthetic division, both of which confirmed that 3x - 2 is indeed a factor when k = 10.

This problem showcases the power and elegance of the Factor Theorem in solving polynomial-related problems. Understanding and applying such theorems is crucial for students and professionals in mathematics, engineering, and related fields.

Polynomials, with their inherent structure and properties, serve as a cornerstone in mathematical analysis and applications. They appear in various contexts, from modeling physical phenomena to designing algorithms in computer science. Mastering polynomial concepts, such as factorization, root finding, and the Factor Theorem, equips individuals with essential tools for problem-solving and critical thinking.

In summary, this exploration not only solved a specific problem but also reinforced the understanding of fundamental algebraic principles. The interplay between factors, roots, and polynomial coefficients is a recurring theme in mathematics, and this article has shed light on how these concepts intertwine to provide solutions to seemingly complex problems.