Dividing Polynomials How To Solve (x³ - X² - 11x + 3) / (x + 3)
Polynomial division might seem daunting at first, but with a clear understanding of the methods involved, it becomes a manageable task. This article provides a detailed walkthrough of how to solve the division problem (x³ - x² - 11x + 3) / (x + 3). We'll explore two common techniques: long division and synthetic division. By the end of this guide, you'll be well-equipped to tackle similar polynomial division problems.
Understanding Polynomial Division
Before diving into the solution, it's crucial to grasp the concept of polynomial division. Just as we divide numbers, we can also divide polynomials. The goal is to find the quotient and remainder when one polynomial is divided by another. In our case, we are dividing the cubic polynomial (x³ - x² - 11x + 3) by the linear polynomial (x + 3). The result will be a polynomial of a lower degree, plus a possible remainder.
Polynomial division is a fundamental operation in algebra, essential for simplifying expressions, solving equations, and understanding the behavior of polynomial functions. Whether you're a student learning algebra or someone brushing up on your math skills, mastering polynomial division is a valuable asset.
Method 1 Long Division
Long division is a method that mirrors the traditional long division you learned in arithmetic, but it is applied to polynomials. It involves systematically dividing the dividend (x³ - x² - 11x + 3) by the divisor (x + 3) step by step.
Setting Up the Long Division
First, we set up the long division problem. Write the dividend (x³ - x² - 11x + 3) inside the division symbol and the divisor (x + 3) outside. Ensure that the terms of the dividend are written in descending order of their exponents.
____________________
x + 3 | x³ - x² - 11x + 3
Step-by-Step Process
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Divide the First Terms: Divide the first term of the dividend (x³) by the first term of the divisor (x). This gives us x².
x²
x + 3 | x³ - x² - 11x + 3 ```
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Multiply: Multiply the quotient term (x²) by the entire divisor (x + 3). This results in x³ + 3x².
x²
x + 3 | x³ - x² - 11x + 3 x³ + 3x² ```
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Subtract: Subtract the result (x³ + 3x²) from the corresponding terms of the dividend (x³ - x²). This yields -4x².
x²
x + 3 | x³ - x² - 11x + 3 -(x³ + 3x²) __________ -4x² ```
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Bring Down: Bring down the next term from the dividend (-11x).
x²
x + 3 | x³ - x² - 11x + 3 -(x³ + 3x²) __________ -4x² - 11x ```
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Repeat: Repeat the process. Divide the new first term (-4x²) by the first term of the divisor (x). This gives us -4x.
x² - 4x
x + 3 | x³ - x² - 11x + 3 -(x³ + 3x²) __________ -4x² - 11x ```
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Multiply: Multiply -4x by the divisor (x + 3), resulting in -4x² - 12x.
x² - 4x
x + 3 | x³ - x² - 11x + 3 -(x³ + 3x²) __________ -4x² - 11x -4x² - 12x ```
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Subtract: Subtract (-4x² - 12x) from (-4x² - 11x), which results in x.
x² - 4x
x + 3 | x³ - x² - 11x + 3 -(x³ + 3x²) __________ -4x² - 11x -(-4x² - 12x) __________ x ```
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Bring Down: Bring down the last term from the dividend (+3).
x² - 4x
x + 3 | x³ - x² - 11x + 3 -(x³ + 3x²) __________ -4x² - 11x -(-4x² - 12x) __________ x + 3 ```
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Repeat: Divide x by x, which gives us 1.
x² - 4x + 1
x + 3 | x³ - x² - 11x + 3 -(x³ + 3x²) __________ -4x² - 11x -(-4x² - 12x) __________ x + 3 ```
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Multiply: Multiply 1 by the divisor (x + 3), resulting in x + 3.
x² - 4x + 1
x + 3 | x³ - x² - 11x + 3 -(x³ + 3x²) __________ -4x² - 11x -(-4x² - 12x) __________ x + 3 x + 3 ```
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Subtract: Subtract (x + 3) from (x + 3), which results in 0. This means there is no remainder.
x² - 4x + 1
x + 3 | x³ - x² - 11x + 3 -(x³ + 3x²) __________ -4x² - 11x -(-4x² - 12x) __________ x + 3 -(x + 3) __________ 0 ```
Result of Long Division
The quotient is x² - 4x + 1, and the remainder is 0. Therefore, (x³ - x² - 11x + 3) / (x + 3) = x² - 4x + 1.
Method 2 Synthetic Division
Synthetic division is a shorthand method of dividing polynomials, particularly useful when dividing by a linear divisor of the form (x - c). It's a more streamlined approach compared to long division, reducing the amount of writing and potential for errors.
Setting Up Synthetic Division
To set up synthetic division, first identify the value of 'c' from the divisor (x + 3). In this case, since we have (x + 3), c = -3. Write 'c' to the left, and then write the coefficients of the dividend (x³ - x² - 11x + 3) across the top row.
-3 | 1 -1 -11 3
|__________________
Step-by-Step Process
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Bring Down: Bring down the first coefficient (1) to the bottom row.
-3 | 1 -1 -11 3 |__________________ 1 ```
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Multiply and Add: Multiply the number you brought down (1) by 'c' (-3), and write the result (-3) under the next coefficient (-1). Add these two numbers (-1 and -3) to get -4.
-3 | 1 -1 -11 3 | -3 |__________________ 1 -4 ```
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Repeat: Repeat the process. Multiply -4 by -3 to get 12, and write it under -11. Add -11 and 12 to get 1.
-3 | 1 -1 -11 3 | -3 12 |__________________ 1 -4 1 ```
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Final Step: Multiply 1 by -3 to get -3, and write it under 3. Add 3 and -3 to get 0.
-3 | 1 -1 -11 3 | -3 12 -3 |__________________ 1 -4 1 0 ```
Interpreting the Result
The numbers in the bottom row, excluding the last one, are the coefficients of the quotient. The last number is the remainder. In this case, we have 1, -4, and 1, which correspond to the coefficients of x², -4x, and 1. The last number, 0, is the remainder.
Therefore, the quotient is x² - 4x + 1, and the remainder is 0. This confirms that (x³ - x² - 11x + 3) / (x + 3) = x² - 4x + 1.
Choosing the Right Method
Both long division and synthetic division are effective methods for dividing polynomials, but they are suited for different situations. Long division is a more general method that works for any polynomial divisor. It's particularly useful when the divisor is not a simple linear expression. Synthetic division, on the other hand, is quicker and more efficient when dividing by a linear divisor of the form (x - c). It simplifies the process and reduces the chance of making errors.
For the problem (x³ - x² - 11x + 3) / (x + 3), synthetic division is the more efficient choice due to the linear divisor (x + 3). However, understanding long division provides a solid foundation for more complex division problems.
Verification
To verify our result, we can multiply the quotient (x² - 4x + 1) by the divisor (x + 3) and check if it equals the dividend (x³ - x² - 11x + 3).
(x² - 4x + 1) * (x + 3) = x³ - 4x² + x + 3x² - 12x + 3 = x³ - x² - 11x + 3
This confirms that our division is correct.
Common Mistakes to Avoid
When performing polynomial division, it’s easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:
- Missing Terms: Ensure that all terms in the dividend are accounted for, including terms with a coefficient of 0. For example, if dividing x⁴ + 1 by x - 1, rewrite x⁴ + 1 as x⁴ + 0x³ + 0x² + 0x + 1.
- Sign Errors: Pay close attention to signs, especially during subtraction steps in long division and the multiplication steps in synthetic division. A simple sign error can lead to an incorrect result.
- Incorrect Multiplication: Double-check your multiplication steps to avoid errors. In synthetic division, multiplying the wrong numbers can throw off the entire process.
- Misinterpreting Results: Make sure you understand how to interpret the results of synthetic division. The numbers in the bottom row represent the coefficients of the quotient, and the last number is the remainder.
Applications of Polynomial Division
Polynomial division isn't just an academic exercise; it has practical applications in various fields. Here are a few examples:
- Factoring Polynomials: If dividing a polynomial by (x - c) results in a remainder of 0, then (x - c) is a factor of the polynomial. This is a useful technique for factoring higher-degree polynomials.
- Solving Polynomial Equations: Polynomial division can help simplify polynomial equations, making them easier to solve. By dividing out known factors, you can reduce the degree of the polynomial and find its roots.
- Graphing Polynomial Functions: Understanding polynomial division can help you analyze the behavior of polynomial functions, such as finding asymptotes and intercepts.
- Engineering and Physics: Polynomials are used to model various phenomena in engineering and physics. Polynomial division can be used to simplify these models and solve related problems.
Conclusion
In conclusion, solving the division problem (x³ - x² - 11x + 3) / (x + 3) yields the quotient x² - 4x + 1. We demonstrated two methods: long division and synthetic division, both of which led to the same result. Long division provides a detailed, step-by-step approach suitable for any polynomial division, while synthetic division offers a quicker, more efficient method for linear divisors. By understanding and practicing these techniques, you can confidently tackle polynomial division problems in various contexts. Remember to avoid common mistakes and verify your results to ensure accuracy. Whether you're working on homework assignments or applying these skills in real-world scenarios, mastering polynomial division is a valuable mathematical tool.
The correct answer to the division problem is A. x² - 4x + 1.
