Synthetic Division Remainder Theorem Polynomial P(x) = X³ - 28x - 48

Jul 8, 2025 by Jeany 69 views

Unveiling the Remainder Theorem Synthetic Division and the Polynomial P(x) = x³ - 28x - 48

In the realm of polynomial algebra, synthetic division stands out as a powerful tool for efficiently dividing a polynomial by a linear divisor of the form x - c. This method not only simplifies the division process but also provides valuable insights into the polynomial's structure, particularly the remainder upon division. In this article, we delve into the application of synthetic division to determine the remainder when the polynomial P(x) = x³ - 28x - 48 is divided by x + 4. We will explore the steps involved in synthetic division, interpret the results, and connect them to the Remainder Theorem, a fundamental concept in polynomial algebra. Understanding these concepts will empower you to tackle similar problems with confidence and gain a deeper appreciation for the elegance of polynomial division.

Understanding Synthetic Division

Before we embark on the specific problem at hand, let's solidify our understanding of synthetic division. Synthetic division is a streamlined method for dividing a polynomial by a linear expression of the form x - c. It's a shorthand version of polynomial long division, but it's often faster and less prone to errors, especially when dealing with higher-degree polynomials. The key to synthetic division lies in focusing on the coefficients of the polynomial and the constant term of the divisor.

At its core, synthetic division leverages the Remainder Theorem, which states that when a polynomial P(x) is divided by x - c, the remainder is equal to P(c). This theorem provides a direct link between the value of a polynomial at a specific point and the remainder obtained upon division by a linear factor. Synthetic division provides an efficient way to calculate this remainder without performing the full polynomial long division.

To illustrate, let's consider the general case of dividing a polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ by x - c. The setup for synthetic division involves writing the coefficients of the polynomial in a row, followed by the value of c (the root of the divisor) to the left. The process then involves a series of multiplications and additions, systematically reducing the polynomial's degree and ultimately revealing the remainder.

By mastering the steps of synthetic division, you gain a powerful tool for polynomial manipulation. This method not only simplifies division but also provides a pathway to factor polynomials, find roots, and solve polynomial equations. The efficiency and elegance of synthetic division make it an indispensable technique in algebra and beyond.

Setting up the Synthetic Division for P(x) = x³ - 28x - 48 Divided by x + 4

Now, let's apply our knowledge of synthetic division to the given problem. We have the polynomial P(x) = x³ - 28x - 48 and the divisor x + 4. Our goal is to find the remainder when P(x) is divided by x + 4 using synthetic division.

The first crucial step is to identify the value of c in the divisor x - c. Since our divisor is x + 4, we can rewrite it as x - (-4). Therefore, c = -4. This value will be the cornerstone of our synthetic division process.

Next, we need to prepare the coefficients of the polynomial P(x). Notice that P(x) has terms for x³ and x, but the x² term is missing. When setting up synthetic division, it's essential to include a placeholder of 0 for any missing terms. So, we rewrite P(x) conceptually as 1x³ + 0x² - 28x - 48. This ensures that the coefficients are aligned correctly during the division process.

Now, we arrange the coefficients and the value of c in the synthetic division format. We write the coefficients (1, 0, -28, -48) in a row, leaving space below them. To the left, we write c = -4. This setup provides a visual framework for the iterative steps of synthetic division.

The setup looks like this:

-4 | 1 0 -28 -48
   |________________________

With the setup complete, we're ready to embark on the core steps of synthetic division. We'll bring down the first coefficient, perform the multiplication and addition steps, and ultimately arrive at the remainder. The careful setup ensures that we maintain the correct order and alignment, leading to an accurate result.

Performing the Synthetic Division

With the synthetic division set up correctly, we can now proceed with the iterative steps that will lead us to the remainder. The process involves a series of bringing down, multiplying, and adding, which systematically reduces the polynomial's degree and isolates the remainder.

Bring Down the First Coefficient: The first step is to bring down the leading coefficient of the polynomial, which in this case is 1. We write this 1 below the line, directly below the original coefficient.
```
-4 | 1 0 -28 -48
   |________________________
   1
```
Multiply and Add: Now, we multiply the value of c (-4) by the number we just brought down (1), which gives us -4. We write this result (-4) below the next coefficient (0).
```
-4 | 1 0 -28 -48
   |    -4
   |________________________
   1
```
Next, we add the numbers in the second column (0 and -4), which gives us -4. We write this sum (-4) below the line.
```
-4 | 1 0 -28 -48
   |    -4
   |________________________
   1 -4
```
Repeat: We repeat the multiply and add process for the remaining columns. Multiply -4 (the value of c) by -4 (the last number written below the line), which gives us 16. Write 16 below the next coefficient (-28).
```
-4 | 1 0 -28 -48
   |    -4 16
   |________________________
   1 -4
```
Add the numbers in the third column (-28 and 16), which gives us -12. Write -12 below the line.
```
-4 | 1 0 -28 -48
   |    -4 16
   |________________________
   1 -4 -12
```
Finally, multiply -4 (the value of c) by -12 (the last number written below the line), which gives us 48. Write 48 below the last coefficient (-48).
```
-4 | 1 0 -28 -48
   |    -4 16 48
   |________________________
   1 -4 -12
```
Add the numbers in the last column (-48 and 48), which gives us 0. Write 0 below the line.
```
-4 | 1 0 -28 -48
   |    -4 16 48
   |________________________
   1 -4 -12 0
```

With the synthetic division complete, we've obtained the crucial numbers that will reveal the remainder and the quotient. The last number below the line (0) represents the remainder, while the other numbers (1, -4, -12) represent the coefficients of the quotient.

Interpreting the Result and Finding the Remainder

Having completed the synthetic division, we now arrive at the crucial step of interpreting the results. The numbers below the line hold the key to understanding the quotient and, most importantly, the remainder when P(x) = x³ - 28x - 48 is divided by x + 4.

As we observed in the previous section, the last number below the line is 0. This number represents the remainder of the division. Therefore, when P(x) is divided by x + 4, the remainder is 0.

This result has significant implications. A remainder of 0 tells us that x + 4 divides P(x) evenly. In other words, x + 4 is a factor of P(x). This connection between the remainder and factors is a cornerstone of polynomial algebra.

Furthermore, the other numbers below the line (1, -4, -12) represent the coefficients of the quotient polynomial. Since we started with a cubic polynomial (x³) and divided by a linear factor (x + 4), the quotient will be a quadratic polynomial. The coefficients 1, -4, and -12 correspond to the terms x², -4x, and -12, respectively. Thus, the quotient is x² - 4x - 12.

In summary, the synthetic division has revealed that when P(x) = x³ - 28x - 48 is divided by x + 4, the remainder is 0, and the quotient is x² - 4x - 12. The remainder of 0 confirms that x + 4 is a factor of P(x), and the quotient provides the other factor that, when multiplied by x + 4, will result in P(x).

Connecting to the Remainder Theorem

The result we obtained through synthetic division, a remainder of 0, is beautifully aligned with the Remainder Theorem. This theorem provides a direct link between the value of a polynomial at a specific point and the remainder obtained upon division by a linear factor.

The Remainder Theorem states that if a polynomial P(x) is divided by x - c, then the remainder is P(c). In our case, we divided P(x) = x³ - 28x - 48 by x + 4, which can be written as x - (-4). Therefore, c = -4.

According to the Remainder Theorem, the remainder should be P(-4). Let's calculate P(-4):

P(-4) = (-4)³ - 28(-4) - 48 P(-4) = -64 + 112 - 48 P(-4) = 0

As we can see, P(-4) = 0, which is exactly the remainder we obtained through synthetic division. This confirms the Remainder Theorem and demonstrates the consistency between the two approaches. The Remainder Theorem provides a theoretical foundation for synthetic division, showing why this efficient method works.

The connection to the Remainder Theorem provides a deeper understanding of the significance of the remainder. A remainder of 0 not only indicates that the divisor is a factor but also that the value c is a root (or zero) of the polynomial. In our case, since P(-4) = 0, we know that -4 is a root of P(x). This link between remainders, factors, and roots is a fundamental concept in polynomial algebra, and the Remainder Theorem provides a powerful tool for exploring these relationships.

Conclusion

In this exploration, we've successfully employed synthetic division to determine the remainder when the polynomial P(x) = x³ - 28x - 48 is divided by x + 4. We meticulously followed the steps of synthetic division, setting up the problem correctly, performing the iterative multiplications and additions, and ultimately arriving at the remainder of 0.

This result signifies that x + 4 divides P(x) evenly, making it a factor of the polynomial. We further connected this finding to the Remainder Theorem, which elegantly confirms that the remainder is equal to P(-4), which we calculated to be 0. This consistency between synthetic division and the Remainder Theorem reinforces the power and elegance of these concepts in polynomial algebra.

Beyond finding the remainder, the synthetic division also provided us with the quotient polynomial, x² - 4x - 12. This quotient represents the other factor that, when multiplied by x + 4, yields the original polynomial P(x). The ability to find both the remainder and the quotient makes synthetic division a valuable tool for polynomial factorization and root finding.

The application of synthetic division and the understanding of the Remainder Theorem not only solve specific problems but also provide a deeper appreciation for the structure and behavior of polynomials. These techniques are essential tools in algebra, calculus, and various fields that rely on mathematical modeling.

By mastering synthetic division and the Remainder Theorem, you gain a significant advantage in tackling polynomial problems. You can efficiently determine remainders, identify factors, and find roots, all of which are crucial skills in mathematical problem-solving and analysis.