Polynomial Division Explained Finding The Quotient Of (x³ + 3x² + 5x + 3) ÷ (x + 1)

Jul 10, 2025 by ADMIN 84 views

Understanding Polynomial Division: Finding the Quotient of (x³ + 3x² + 5x + 3) ÷ (x + 1)

Introduction

In the realm of algebra, polynomial division is a fundamental operation that allows us to break down complex expressions into simpler forms. This article delves into the process of dividing the polynomial x³ + 3x² + 5x + 3 by the binomial x + 1, aiming to find the quotient. We will explore the steps involved in polynomial long division and arrive at the correct answer, enhancing your understanding of this crucial algebraic concept. This article not only provides the solution but also helps to illustrate the underlying principles and techniques of polynomial division, which can be applied to various mathematical problems. The ability to confidently perform polynomial division is essential for solving equations, simplifying expressions, and understanding more advanced topics in algebra and calculus.

Polynomial Long Division: A Step-by-Step Guide

Polynomial long division is a systematic method for dividing one polynomial by another polynomial of equal or lower degree. It is similar to the long division method used for dividing numbers. To find the quotient of (x³ + 3x² + 5x + 3) ÷ (x + 1), we will use polynomial long division. The long division method is a step-by-step process that breaks down the division into manageable parts, ensuring accuracy and clarity. By understanding each step, you can confidently tackle various polynomial division problems. This method is particularly useful when factoring polynomials or solving equations involving polynomial expressions. Mastering polynomial long division is a valuable skill for any student studying algebra and beyond.

Set up the division: Write the dividend (x³ + 3x² + 5x + 3) inside the division symbol and the divisor (x + 1) outside.
```
        _________
```

x + 1 | x³ + 3x² + 5x + 3 ```

Divide the first term: Divide the first term of the dividend (x³) by the first term of the divisor (x). The result is x². Write x² above the division symbol.
```
        x²________
```

x + 1 | x³ + 3x² + 5x + 3 ```

Multiply: Multiply the quotient term (x²) by the entire divisor (x + 1). This gives x³ + x².
```
        x²________
```

x + 1 | x³ + 3x² + 5x + 3 x³ + x² ```

Subtract: Subtract the result from the corresponding terms of the dividend.
```
        x²________
```

x + 1 | x³ + 3x² + 5x + 3 -(x³ + x²) --------- 2x² ```

Bring down the next term: Bring down the next term from the dividend (5x).
```
        x²________
```

x + 1 | x³ + 3x² + 5x + 3 -(x³ + x²) --------- 2x² + 5x ```

Repeat the process: Divide the first term of the new dividend (2x²) by the first term of the divisor (x). The result is 2x. Write +2x next to x² above the division symbol.
```
        x² + 2x_____
```

x + 1 | x³ + 3x² + 5x + 3 -(x³ + x²) --------- 2x² + 5x ```

Multiply: Multiply the new quotient term (2x) by the entire divisor (x + 1). This gives 2x² + 2x.
```
        x² + 2x_____
```

x + 1 | x³ + 3x² + 5x + 3 -(x³ + x²) --------- 2x² + 5x 2x² + 2x ```

Subtract: Subtract the result from the corresponding terms of the current dividend.
```
        x² + 2x_____
```

x + 1 | x³ + 3x² + 5x + 3 -(x³ + x²) --------- 2x² + 5x -(2x² + 2x) --------- 3x ```

Bring down the next term: Bring down the next term from the dividend (3).
```
        x² + 2x_____
```

x + 1 | x³ + 3x² + 5x + 3 -(x³ + x²) --------- 2x² + 5x -(2x² + 2x) --------- 3x + 3 ```

Repeat the process again: Divide the first term of the new dividend (3x) by the first term of the divisor (x). The result is 3. Write +3 next to x² + 2x above the division symbol.
```
        x² + 2x + 3
```

x + 1 | x³ + 3x² + 5x + 3 -(x³ + x²) --------- 2x² + 5x -(2x² + 2x) --------- 3x + 3 ```

Multiply: Multiply the new quotient term (3) by the entire divisor (x + 1). This gives 3x + 3.
```
        x² + 2x + 3
```

x + 1 | x³ + 3x² + 5x + 3 -(x³ + x²) --------- 2x² + 5x -(2x² + 2x) --------- 3x + 3 3x + 3 ```

Subtract: Subtract the result from the corresponding terms of the current dividend.
```
        x² + 2x + 3
```

x + 1 | x³ + 3x² + 5x + 3 -(x³ + x²) --------- 2x² + 5x -(2x² + 2x) --------- 3x + 3 -(3x + 3) --------- 0 ```

Determine the quotient and remainder: The quotient is x² + 2x + 3, and the remainder is 0.

The Quotient

After performing the polynomial long division, we find that the quotient of (x³ + 3x² + 5x + 3) ÷ (x + 1) is x² + 2x + 3. This means that when the polynomial x³ + 3x² + 5x + 3 is divided by x + 1, the result is the quadratic polynomial x² + 2x + 3. The quotient represents the result of the division, and in this case, it is a polynomial of degree 2. Understanding how to find the quotient in polynomial division is essential for simplifying expressions, solving equations, and further exploring algebraic concepts.

Conclusion

In conclusion, by using polynomial long division, we have successfully determined that the quotient of (x³ + 3x² + 5x + 3) ÷ (x + 1) is x² + 2x + 3. Polynomial division is a vital technique in algebra, and mastering it allows for the simplification and manipulation of polynomial expressions. The step-by-step approach of long division ensures accuracy and a clear understanding of the process. This skill is invaluable for solving more complex problems in mathematics and related fields. The ability to confidently perform polynomial division opens doors to more advanced topics and enhances overall problem-solving capabilities in mathematics. Through practice and understanding, polynomial division becomes an indispensable tool in any mathematical toolkit.

Therefore, the correct answer is:

A. x² + 2x + 3