Polynomial Division: Find The Quotient Of (x³ + 3x² + 5x + 3) ÷ (x + 1)

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Hey guys! Today, we're diving into the world of polynomial division. Specifically, we're going to figure out what happens when we divide the polynomial (x³ + 3x² + 5x + 3) by (x + 1). This might sound intimidating, but trust me, we'll break it down step-by-step. Polynomial division is a fundamental concept in algebra, and mastering it opens doors to solving more complex equations and understanding various mathematical relationships. It's not just about crunching numbers; it's about understanding how polynomials interact and how we can manipulate them. So, let's grab our algebraic toolkits and get started!

Long Division Method

The most common method for dividing polynomials is long division, which is very similar to the long division you learned in elementary school with numbers. Here's how it works:

  1. Set up the division: Write the dividend (x³ + 3x² + 5x + 3) inside the division symbol and the divisor (x + 1) outside.
  2. Divide the first term: Divide the first term of the dividend (x³) by the first term of the divisor (x). This gives you x², which is the first term of the quotient. Write this above the division symbol.
  3. Multiply: Multiply the divisor (x + 1) by the first term of the quotient (x²). This gives you x³ + x².
  4. Subtract: Subtract the result (x³ + x²) from the first two terms of the dividend (x³ + 3x²). This gives you 2x².
  5. Bring down the next term: Bring down the next term from the dividend (+5x) to get 2x² + 5x.
  6. Repeat: Divide the first term of the new expression (2x²) by the first term of the divisor (x). This gives you +2x, which is the next term of the quotient. Write this above the division symbol.
  7. Multiply: Multiply the divisor (x + 1) by the new term of the quotient (2x). This gives you 2x² + 2x.
  8. Subtract: Subtract the result (2x² + 2x) from (2x² + 5x). This gives you 3x.
  9. Bring down the last term: Bring down the last term from the dividend (+3) to get 3x + 3.
  10. Final Repeat: Divide the first term of the new expression (3x) by the first term of the divisor (x). This gives you +3, which is the last term of the quotient. Write this above the division symbol.
  11. Multiply: Multiply the divisor (x + 1) by the new term of the quotient (3). This gives you 3x + 3.
  12. Subtract: Subtract the result (3x + 3) from (3x + 3). This gives you 0.

Since the remainder is 0, the division is exact, and the quotient is x² + 2x + 3.

Synthetic Division Method

Another method, which is often quicker for dividing by a linear divisor like (x + 1), is synthetic division. Here’s how it works:

  1. Set up: Write down the coefficients of the dividend (1, 3, 5, 3). Since we're dividing by (x + 1), we use -1 (the root of x + 1 = 0).
  2. Bring down the first coefficient: Bring down the first coefficient (1).
  3. Multiply and add: Multiply the number you brought down (1) by -1 to get -1. Add this to the next coefficient (3) to get 2.
  4. Repeat: Multiply the result (2) by -1 to get -2. Add this to the next coefficient (5) to get 3.
  5. Final Repeat: Multiply the result (3) by -1 to get -3. Add this to the last coefficient (3) to get 0.

The numbers you get (1, 2, 3) are the coefficients of the quotient, and the last number (0) is the remainder. So, the quotient is x² + 2x + 3.

Synthetic division is a streamlined process that can save you time and effort, especially when dealing with linear divisors. It's a valuable tool to have in your mathematical arsenal, allowing you to quickly and efficiently find the quotient of polynomial divisions.

Verification

To be absolutely sure, let's multiply the quotient (x² + 2x + 3) by the divisor (x + 1):

(x² + 2x + 3) * (x + 1) = x³ + x² + 2x² + 2x + 3x + 3 = x³ + 3x² + 5x + 3

This matches the original dividend, so our quotient is correct!

Verifying your result is a crucial step in any mathematical problem. It's like double-checking your work to ensure you haven't made any mistakes along the way. By multiplying the quotient by the divisor, you can confirm that you arrive back at the original dividend, giving you confidence in your solution. It's a simple yet effective way to avoid errors and ensure accuracy.

Why is Polynomial Division Important?

You might be wondering, why bother with all this polynomial division stuff? Well, it's incredibly useful in many areas of mathematics and engineering:

  • Factoring polynomials: Finding the roots of polynomials becomes much easier.
  • Solving equations: Simplifying complex equations to find solutions.
  • Calculus: Simplifying rational functions for integration and differentiation.
  • Engineering: Modeling and analyzing systems in various fields.

Polynomial division is a cornerstone of algebraic manipulation, providing the tools to simplify complex expressions, solve equations, and understand the behavior of polynomial functions. It's a skill that unlocks doors to advanced mathematical concepts and practical applications in various fields. Mastering polynomial division empowers you to tackle challenging problems and gain a deeper understanding of the mathematical world around you.

Conclusion

So, the quotient of (x³ + 3x² + 5x + 3) ÷ (x + 1) is x² + 2x + 3. The correct answer is C. Whether you prefer long division or synthetic division, the key is to understand the process and practice, practice, practice!

Remember, practice makes perfect. The more you work with polynomial division, the more comfortable and confident you'll become. Don't be afraid to make mistakes along the way; they're valuable learning opportunities. With dedication and perseverance, you'll master this essential skill and unlock a world of mathematical possibilities. Keep practicing, and you'll be amazed at what you can achieve!