Adding Polynomials A Step-by-Step Guide To Simplify (5x³ - 3x² + 7x) + (7x³ + 9x² - 13x)

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In this comprehensive guide, we will delve into the process of adding polynomials, specifically focusing on the expression (5x³ - 3x² + 7x) + (7x³ + 9x² - 13x). Polynomial addition involves combining like terms, which are terms that have the same variable raised to the same power. By systematically identifying and combining these terms, we can simplify the expression and arrive at the final answer. This article will provide a step-by-step breakdown of the process, ensuring a clear understanding of the underlying principles and techniques involved in polynomial addition. Whether you're a student learning algebra or simply seeking to refresh your knowledge, this guide will equip you with the skills to confidently tackle similar problems.

Understanding Polynomials and Like Terms

Before we dive into the specifics of adding the given polynomials, let's first establish a firm understanding of what polynomials are and what constitutes like terms. A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, where the exponents of the variables are non-negative integers. For instance, the expressions 5x³ - 3x² + 7x and 7x³ + 9x² - 13x are both polynomials.

Like terms are terms within a polynomial (or across multiple polynomials being added or subtracted) that have the same variable raised to the same power. The coefficients of like terms can be different, but the variable and its exponent must be identical. For example, in the expression 5x³ - 3x² + 7x, the terms 5x³, -3x², and 7x are all distinct terms. However, when adding polynomials, we look for terms that share the same variable and exponent across the expressions being added.

In the given problem, (5x³ - 3x² + 7x) + (7x³ + 9x² - 13x), we can identify the following like terms:

  • 5x³ and 7x³ (both are cubic terms, with x raised to the power of 3)
  • -3x² and 9x² (both are quadratic terms, with x raised to the power of 2)
  • 7x and -13x (both are linear terms, with x raised to the power of 1)

Recognizing and grouping like terms is the foundational step in adding polynomials. It allows us to combine the coefficients of these terms while maintaining the variable and exponent, effectively simplifying the expression.

Step-by-Step Process of Adding Polynomials

Now that we have a solid understanding of polynomials and like terms, let's walk through the step-by-step process of adding the given polynomials: (5x³ - 3x² + 7x) + (7x³ + 9x² - 13x).

Step 1: Identify Like Terms

As we discussed earlier, the first step is to identify the like terms within the expression. We have already identified them as:

  • 5x³ and 7x³
  • -3x² and 9x²
  • 7x and -13x

Step 2: Group Like Terms

Next, we group the like terms together. This can be done by rearranging the expression to bring the like terms next to each other. This step is primarily for visual clarity and can help prevent errors in the subsequent step.

(5x³ + 7x³) + (-3x² + 9x²) + (7x - 13x)

Notice how we have grouped the cubic terms, the quadratic terms, and the linear terms together using parentheses. This makes it easier to see which terms we will be combining.

Step 3: Combine Like Terms

This is the core step in polynomial addition. We combine like terms by adding their coefficients while keeping the variable and exponent the same. Remember, we are only adding the coefficients, not changing the variables or their powers.

  • For the cubic terms: 5x³ + 7x³ = (5 + 7)x³ = 12x³
  • For the quadratic terms: -3x² + 9x² = (-3 + 9)x² = 6x²
  • For the linear terms: 7x - 13x = (7 - 13)x = -6x

Step 4: Write the Simplified Expression

Finally, we write the simplified expression by combining the results from Step 3. We arrange the terms in descending order of their exponents, which is a standard convention for writing polynomials.

12x³ + 6x² - 6x

Therefore, the simplified form of the expression (5x³ - 3x² + 7x) + (7x³ + 9x² - 13x) is 12x³ + 6x² - 6x.

Examples and Practice Problems

To solidify your understanding of polynomial addition, let's work through a few more examples and practice problems.

Example 1:

Add the polynomials (2x² + 5x - 3) + (4x² - 2x + 1)

  1. Identify Like Terms: 2x² and 4x², 5x and -2x, -3 and 1
  2. Group Like Terms: (2x² + 4x²) + (5x - 2x) + (-3 + 1)
  3. Combine Like Terms: 6x² + 3x - 2

Therefore, the simplified expression is 6x² + 3x - 2.

Example 2:

Add the polynomials (x³ - 4x + 2) + (3x² + x - 5)

  1. Identify Like Terms: x³, 3x², -4x and x, 2 and -5
  2. Group Like Terms: x³ + 3x² + (-4x + x) + (2 - 5)
  3. Combine Like Terms: x³ + 3x² - 3x - 3

Therefore, the simplified expression is x³ + 3x² - 3x - 3.

Practice Problems:

  1. (3x⁴ - 2x² + x) + (x⁴ + 5x² - 3x)
  2. (7x³ + x² - 4) + (2x³ - 6x² + 9)
  3. (4x² - 3x + 1) + (x² + 2x - 5) + (2x² - x + 3)

Try solving these problems yourself, following the step-by-step process we outlined earlier. The answers are provided at the end of this article for you to check your work.

Common Mistakes and How to Avoid Them

Polynomial addition is a relatively straightforward process, but it's easy to make mistakes if you're not careful. Here are some common mistakes and how to avoid them:

  1. Forgetting to Distribute a Negative Sign: When subtracting polynomials, remember to distribute the negative sign to all terms within the parentheses being subtracted. This is crucial for correctly identifying and combining like terms.
  2. Combining Unlike Terms: This is one of the most common mistakes. Only like terms (terms with the same variable and exponent) can be combined. Make sure you carefully identify and group like terms before adding their coefficients.
  3. Incorrectly Adding Coefficients: Double-check your addition of coefficients, especially when dealing with negative numbers. A small arithmetic error can lead to a completely wrong answer.
  4. Not Arranging Terms in Descending Order: While not technically an error, it's standard practice to write polynomials in descending order of their exponents. This makes the polynomial easier to read and compare with others.

By being mindful of these common mistakes and practicing diligently, you can avoid them and improve your accuracy in polynomial addition.

Real-World Applications of Polynomials

Polynomials are not just abstract mathematical concepts; they have numerous applications in various fields, including:

  • Engineering: Polynomials are used to model curves and surfaces in engineering design, such as the shape of a bridge or the aerodynamic profile of an airplane wing.
  • Physics: Polynomials can describe the motion of objects, such as the trajectory of a projectile or the oscillations of a pendulum.
  • Computer Graphics: Polynomials are used to create smooth curves and surfaces in computer graphics and animation.
  • Economics: Polynomial functions can model cost, revenue, and profit in economic analysis.
  • Statistics: Polynomial regression is a statistical technique that uses polynomials to model the relationship between variables.

The ability to manipulate and simplify polynomials, including adding them, is essential for understanding and applying these concepts in real-world scenarios. Mastering polynomial addition is a fundamental building block for more advanced mathematical and scientific studies.

Conclusion

In this comprehensive guide, we have explored the process of adding polynomials, focusing on the example (5x³ - 3x² + 7x) + (7x³ + 9x² - 13x). We learned the importance of identifying and grouping like terms, combining their coefficients, and writing the simplified expression in descending order of exponents. We also discussed common mistakes to avoid and the real-world applications of polynomials.

By understanding the principles and techniques outlined in this article, you are now well-equipped to confidently add polynomials and tackle more complex algebraic problems. Remember to practice regularly and apply your knowledge to various examples and scenarios. With consistent effort, you can master polynomial addition and unlock its potential in diverse fields of study and application.

Answers to Practice Problems:

  1. 4x⁴ + 3x² - 2x
  2. 9x³ - 5x² + 5
  3. 7x² - 2x - 1