Solving Polynomial Division Problems Step-by-Step

Jul 11, 2025 by ADMIN 50 views

Unlocking the Solution to Polynomial Division: A Step-by-Step Guide

#Problem

We are presented with a division problem involving polynomials: $\frac{x^3-2 x^2-14 x+3}{x+3}$ . Our task is to find the quotient when the polynomial $x^3 - 2x^2 - 14x + 3$ is divided by $x + 3$ . This type of problem is a staple in algebra, and understanding how to solve it is crucial for mastering polynomial manipulation. The options provided suggest that the result will be a quadratic expression, and we need to determine the correct one from the choices given:

A. $x^2-7 x+1$ B. $x^2-5 x+1$ C. $x^2-8 x+1$ D. $x^2-6 x+1$

Methods to Solve Polynomial Division

There are primarily two methods to tackle polynomial division problems like this: long division and synthetic division. Both methods systematically break down the division process, but they approach it in slightly different ways. For this particular problem, we will demonstrate both methods to provide a comprehensive understanding. Long division is a method that mirrors the traditional long division process used with numbers, while synthetic division is a shorthand method that is particularly efficient when dividing by a linear factor of the form x - a.

1. Long Division Method

The long division method is a robust technique that can handle division by any polynomial, not just linear ones. It involves setting up the division problem in a similar format to numerical long division and then iteratively dividing, multiplying, subtracting, and bringing down terms. Let's walk through the steps:

Set up the long division: Write the dividend ( $x^3 - 2x^2 - 14x + 3$ ) inside the division symbol and the divisor ( $x + 3$ ) outside.
```
       _____________
```

x + 3 | x^3 - 2x^2 - 14x + 3 ```

Divide the first term of the dividend by the first term of the divisor: $x^3$ divided by $x$ is $x^2$ . Write $x^2$ above the division symbol.
```
       x^2__________
```

x + 3 | x^3 - 2x^2 - 14x + 3 ```

Multiply the quotient term ( $x^2$ ) by the entire divisor ( $x + 3$ ): $x^2 * (x + 3) = x^3 + 3x^2$ . Write this result below the dividend.
```
       x^2__________
```

x + 3 | x^3 - 2x^2 - 14x + 3 x^3 + 3x^2 ```

Subtract the result from the corresponding terms in the dividend: $(x^3 - 2x^2) - (x^3 + 3x^2) = -5x^2$ . Bring down the next term (-14x) from the dividend.
```
       x^2__________
```

x + 3 | x^3 - 2x^2 - 14x + 3 - (x^3 + 3x^2) ------------- -5x^2 - 14x ```

Repeat the process: Divide the first term of the new expression (-5x^2) by the first term of the divisor (x): $-5x^2$ divided by $x$ is $-5x$ . Write $-5x$ above the division symbol.
```
       x^2 - 5x______
```

x + 3 | x^3 - 2x^2 - 14x + 3 - (x^3 + 3x^2) ------------- -5x^2 - 14x ```

Multiply the new quotient term (-5x) by the divisor (x + 3): $-5x * (x + 3) = -5x^2 - 15x$ . Write this result below the -5x^2 - 14x.
```
       x^2 - 5x______
```

x + 3 | x^3 - 2x^2 - 14x + 3 - (x^3 + 3x^2) ------------- -5x^2 - 14x -5x^2 - 15x ```

Subtract the result: $(-5x^2 - 14x) - (-5x^2 - 15x) = x$ . Bring down the next term (+3) from the dividend.
```
       x^2 - 5x______
```

x + 3 | x^3 - 2x^2 - 14x + 3 - (x^3 + 3x^2) ------------- -5x^2 - 14x - (-5x^2 - 15x) ------------- x + 3 ```

Repeat the process one last time: Divide the first term of the new expression (x) by the first term of the divisor (x): $x$ divided by $x$ is $1$ . Write $+1$ above the division symbol.
```
       x^2 - 5x + 1__
```

x + 3 | x^3 - 2x^2 - 14x + 3 - (x^3 + 3x^2) ------------- -5x^2 - 14x - (-5x^2 - 15x) ------------- x + 3 ```

Multiply the new quotient term (1) by the divisor (x + 3): $1 * (x + 3) = x + 3$ . Write this result below the $x + 3$ .
```
       x^2 - 5x + 1__
```

x + 3 | x^3 - 2x^2 - 14x + 3 - (x^3 + 3x^2) ------------- -5x^2 - 14x - (-5x^2 - 15x) ------------- x + 3 x + 3 ```

Subtract the result: $(x + 3) - (x + 3) = 0$ . The remainder is 0.
```
       x^2 - 5x + 1__
```

x + 3 | x^3 - 2x^2 - 14x + 3 - (x^3 + 3x^2) ------------- -5x^2 - 14x - (-5x^2 - 15x) ------------- x + 3 - (x + 3) ------------- 0 ```

Therefore, the quotient is $x^2 - 5x + 1$ .

2. Synthetic Division Method

The synthetic division method provides a more streamlined approach, especially when dividing by a linear expression of the form x - a. This method uses coefficients and a specific value derived from the divisor to perform the division. Here's how it works:

Identify the value of a from the divisor: In our case, the divisor is $x + 3$ , which can be written as $x - (-3)$ . So, a = -3.
Write down the coefficients of the dividend: The coefficients of $x^3 - 2x^2 - 14x + 3$ are 1, -2, -14, and 3.
Set up the synthetic division table: Write a (-3) to the left, and the coefficients to the right.
```
-3 | 1  -2  -14   3
   |_________________
```
Bring down the first coefficient: Bring down the 1 to the bottom row.
```
-3 | 1  -2  -14   3
   |_________________
      1
```
Multiply the value just brought down by a: Multiply 1 by -3 to get -3. Write this under the next coefficient (-2).
```
-3 | 1  -2  -14   3
   |     -3
   |_________________
      1
```
Add the numbers in the column: Add -2 and -3 to get -5. Write this in the bottom row.
```
-3 | 1  -2  -14   3
   |     -3
   |_________________
      1  -5
```
Repeat the process: Multiply -5 by -3 to get 15. Write this under the next coefficient (-14).
```
-3 | 1  -2  -14   3
   |     -3  15
   |_________________
      1  -5
```
Add the numbers in the column: Add -14 and 15 to get 1. Write this in the bottom row.
```
-3 | 1  -2  -14   3
   |     -3  15
   |_________________
      1  -5   1
```
Repeat the process one last time: Multiply 1 by -3 to get -3. Write this under the last coefficient (3).
```
-3 | 1  -2  -14   3
   |     -3  15  -3
   |_________________
      1  -5   1
```

Add the numbers in the column: Add 3 and -3 to get 0. Write this in the bottom row.

-3 | 1  -2  -14   3
   |     -3  15  -3
   |_________________
      1  -5   1   0

Interpret the result: The numbers in the bottom row (except the last one) are the coefficients of the quotient, and the last number is the remainder. In this case, the coefficients are 1, -5, and 1, and the remainder is 0. This means the quotient is $1x^2 - 5x + 1$ , or $x^2 - 5x + 1$ .

Conclusion

Both the long division and synthetic division methods lead us to the same solution. By performing either method correctly, we find that the result of dividing $x^3 - 2x^2 - 14x + 3$ by $x + 3$ is $x^2 - 5x + 1$ . Therefore, the correct answer is:

B. $x^2 - 5x + 1$

Understanding polynomial division is a fundamental skill in algebra. Whether you prefer the methodical approach of long division or the efficiency of synthetic division, mastering these techniques will empower you to solve a wide range of problems involving polynomial expressions.