Finding K When 2x³ + 9x² - 7x + K Is Divisible By 2x + 3
Introduction
In the realm of polynomial algebra, determining the divisibility of one polynomial by another is a fundamental concept. When a polynomial, say P(x), is exactly divisible by another polynomial, say D(x), it implies that the remainder upon division is zero. This principle is crucial in various mathematical applications, including finding roots of polynomials, simplifying algebraic expressions, and solving equations. In this article, we will delve into a specific problem where we need to find the value of k such that the polynomial 2x³ + 9x² - 7x + k is exactly divisible by 2x + 3. We will employ the powerful technique of synthetic division to achieve this. Synthetic division offers a streamlined approach to polynomial division, especially when the divisor is a linear expression. It simplifies the process by focusing on the coefficients of the polynomials, making it less cumbersome than traditional long division. Before diving into the solution, it's essential to grasp the underlying principles of polynomial division and the concept of remainders. When a polynomial P(x) is divided by a divisor D(x), we obtain a quotient Q(x) and a remainder R(x). The relationship can be expressed as P(x) = D(x) * Q(x) + R(x). If P(x) is exactly divisible by D(x), then the remainder R(x) is zero. This condition forms the basis for solving our problem. Synthetic division, in particular, is a shortcut method for dividing a polynomial by a linear divisor of the form x - a. It leverages the coefficients of the polynomial and the root of the divisor to efficiently compute the quotient and remainder. By understanding the mechanics of synthetic division, we can systematically determine the value of k that makes the given polynomial divisible by 2x + 3.
Understanding the Problem
Our primary objective is to find the value of k in the polynomial 2x³ + 9x² - 7x + k such that the polynomial is exactly divisible by 2x + 3. Exact divisibility implies that when we divide the given cubic polynomial by the linear expression 2x + 3, the remainder should be zero. This condition is crucial because it allows us to set up an equation involving k, which we can then solve. To effectively tackle this problem, we will utilize the method of synthetic division. However, before we jump into the synthetic division process, let's address a crucial step: transforming the divisor into the standard form required for synthetic division. Synthetic division is typically applied when the divisor is in the form x - a, where a is a constant. Our divisor, however, is 2x + 3. To convert it into the required form, we need to find the value of x that makes the divisor equal to zero. Setting 2x + 3 = 0, we solve for x and find that x = -3/2. This value, -3/2, will be the key element we use in our synthetic division setup. Now that we have the divisor in the appropriate form for synthetic division, we can proceed with the process. Synthetic division involves a systematic arrangement of the coefficients of the dividend polynomial and the root of the divisor. By following the steps of synthetic division, we can efficiently compute the quotient and the remainder. The remainder, in particular, is of utmost importance to us because we know that it must be zero for the polynomial to be exactly divisible. Therefore, by setting the remainder equal to zero, we can form an equation involving k and solve for its value. This entire process hinges on understanding the relationship between polynomial division, remainders, and the specific form required for synthetic division. By carefully applying these concepts, we can confidently find the value of k that satisfies the given condition.
Applying Synthetic Division
Now, let's apply the method of synthetic division to our problem. We have the polynomial 2x³ + 9x² - 7x + k and the divisor 2x + 3. As we discussed earlier, we first need to find the value of x that makes the divisor zero, which we found to be x = -3/2. This value will be used in our synthetic division setup. The coefficients of the polynomial are 2, 9, -7, and k. We arrange these coefficients in a row, and the value -3/2 goes to the left, as follows:
-3/2 | 2 9 -7 k
|________________________
The first step in synthetic division is to bring down the leading coefficient, which is 2, below the line:
-3/2 | 2 9 -7 k
|________________________
2
Next, we multiply -3/2 by the number we just brought down (which is 2), and write the result above the line under the next coefficient (which is 9). So, (-3/2) * 2 = -3:
-3/2 | 2 9 -7 k
| -3
|________________________
2
Now, we add the numbers in the second column: 9 + (-3) = 6, and write the result below the line:
-3/2 | 2 9 -7 k
| -3
|________________________
2 6
We repeat this process for the remaining coefficients. Multiply -3/2 by 6, which gives us -9, and write it above the line under the next coefficient (which is -7):
-3/2 | 2 9 -7 k
| -3 -9
|________________________
2 6
Add the numbers in the third column: -7 + (-9) = -16, and write the result below the line:
-3/2 | 2 9 -7 k
| -3 -9
|________________________
2 6 -16
Finally, multiply -3/2 by -16, which gives us 24, and write it above the line under the last coefficient (which is k):
-3/2 | 2 9 -7 k
| -3 -9 24
|________________________
2 6 -16
Add the numbers in the last column: k + 24, and write the result below the line. This result represents the remainder of the division:
-3/2 | 2 9 -7 k
| -3 -9 24
|________________________
2 6 -16 k+24
The last number below the line, k + 24, is the remainder. For the polynomial to be exactly divisible by 2x + 3, the remainder must be zero. This gives us the equation k + 24 = 0.
Solving for k
In the previous section, we applied synthetic division and found that the remainder when 2x³ + 9x² - 7x + k is divided by 2x + 3 is k + 24. For the polynomial to be exactly divisible, the remainder must be zero. Therefore, we have the equation:
k + 24 = 0
To solve for k, we need to isolate k on one side of the equation. We can do this by subtracting 24 from both sides of the equation:
k + 24 - 24 = 0 - 24
This simplifies to:
k = -24
Thus, the value of k that makes the polynomial 2x³ + 9x² - 7x + k exactly divisible by 2x + 3 is -24. This means that if we substitute -24 for k in the original polynomial, the resulting polynomial will be perfectly divisible by 2x + 3, leaving no remainder.
To verify our solution, we can substitute k = -24 back into the original polynomial and perform the division. The polynomial becomes:
2x³ + 9x² - 7x - 24
We can divide this polynomial by 2x + 3 using either synthetic division or long division to confirm that the remainder is indeed zero. If the remainder is zero, it validates our solution that k = -24 is the correct value.
In conclusion, by applying synthetic division and setting the remainder equal to zero, we have successfully determined the value of k that ensures the polynomial 2x³ + 9x² - 7x + k is exactly divisible by 2x + 3. The value of k is -24. This solution demonstrates the power of synthetic division as a tool for solving polynomial division problems and finding unknown coefficients.
Conclusion
In this comprehensive exploration, we successfully determined the value of k for which the polynomial 2x³ + 9x² - 7x + k is exactly divisible by 2x + 3. By leveraging the technique of synthetic division, we systematically navigated through the division process, ultimately arriving at the solution k = -24. Our journey began with a clear understanding of the problem, emphasizing the significance of exact divisibility and its implication of a zero remainder. We then delved into the mechanics of synthetic division, meticulously outlining each step from setting up the division to computing the remainder. The crucial step involved recognizing the need to transform the divisor into the standard form required for synthetic division, which led us to find the value x = -3/2. This value served as the cornerstone of our synthetic division process. As we meticulously applied synthetic division, we carefully tracked the coefficients and performed the necessary multiplications and additions. The culmination of this process was the determination of the remainder, expressed as k + 24. Recognizing that the remainder must be zero for exact divisibility, we set up the equation k + 24 = 0 and solved for k, arriving at the solution k = -24. This result signifies that when k is equal to -24, the polynomial 2x³ + 9x² - 7x + k is perfectly divisible by 2x + 3, leaving no remainder. The power of synthetic division lies in its efficiency and simplicity. It provides a streamlined approach to polynomial division, particularly when the divisor is a linear expression. By focusing on the coefficients and the root of the divisor, synthetic division simplifies the process and reduces the likelihood of errors. Furthermore, this problem highlights the interconnectedness of various concepts in polynomial algebra. The relationship between divisibility, remainders, and the coefficients of polynomials is crucial for solving such problems. By mastering these concepts, one can confidently tackle a wide range of algebraic challenges. In conclusion, our exploration demonstrates the elegance and effectiveness of synthetic division in solving polynomial divisibility problems. The solution k = -24 not only answers the specific question but also reinforces the importance of understanding the underlying principles of polynomial algebra.