How To Simplify (2x² + 3x - 3)(3x² + 2x + 3) A Step-by-Step Guide

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In the realm of algebra, simplifying polynomial expressions is a fundamental skill. Polynomials, which are expressions consisting of variables and coefficients, often appear in various mathematical contexts, from solving equations to modeling real-world phenomena. One common task involves expanding the product of two polynomials, which requires careful application of the distributive property. In this comprehensive guide, we will delve into the process of simplifying the expression (2x² + 3x - 3)(3x² + 2x + 3), providing a step-by-step approach that will empower you to tackle similar problems with confidence.

Understanding Polynomial Multiplication

Before we embark on the simplification process, let's lay the groundwork by understanding the principles of polynomial multiplication. When multiplying two polynomials, we essentially distribute each term of the first polynomial across every term of the second polynomial. This process ensures that every possible combination of terms is accounted for, leading to the expanded form of the expression. The distributive property, which states that a(b + c) = ab + ac, serves as the cornerstone of this operation. Applying this property systematically allows us to break down the complex multiplication into a series of simpler multiplications, ultimately paving the way for simplification.

Step-by-Step Simplification of (2x² + 3x - 3)(3x² + 2x + 3)

Now, let's embark on the journey of simplifying the expression (2x² + 3x - 3)(3x² + 2x + 3). We'll break down the process into manageable steps, providing explanations and insights along the way.

Step 1: Distribute the First Term (2x²)

Our initial step involves distributing the first term of the first polynomial, which is 2x², across each term of the second polynomial. This yields:

2x² * (3x² + 2x + 3) = (2x² * 3x²) + (2x² * 2x) + (2x² * 3) = 6x⁴ + 4x³ + 6x²

Step 2: Distribute the Second Term (3x)

Next, we distribute the second term of the first polynomial, which is 3x, across each term of the second polynomial. This gives us:

3x * (3x² + 2x + 3) = (3x * 3x²) + (3x * 2x) + (3x * 3) = 9x³ + 6x² + 9x

Step 3: Distribute the Third Term (-3)

Now, we distribute the third term of the first polynomial, which is -3, across each term of the second polynomial. This results in:

-3 * (3x² + 2x + 3) = (-3 * 3x²) + (-3 * 2x) + (-3 * 3) = -9x² - 6x - 9

Step 4: Combine the Results

Having distributed each term of the first polynomial across the second polynomial, we now have three sets of terms. To complete the simplification, we need to combine these terms:

(6x⁴ + 4x³ + 6x²) + (9x³ + 6x² + 9x) + (-9x² - 6x - 9)

Step 5: Identify and Combine Like Terms

Like terms are terms that have the same variable raised to the same power. To simplify the expression further, we identify and combine these like terms. In this case, we have:

  • x⁴ terms: 6x⁴ (only one term)
  • x³ terms: 4x³ + 9x³ = 13x³
  • x² terms: 6x² + 6x² - 9x² = 3x²
  • x terms: 9x - 6x = 3x
  • Constant terms: -9 (only one term)

Step 6: Write the Simplified Expression

Finally, we combine the simplified like terms to obtain the final expression:

6x⁴ + 13x³ + 3x² + 3x - 9

Therefore, the simplified form of (2x² + 3x - 3)(3x² + 2x + 3) is 6x⁴ + 13x³ + 3x² + 3x - 9. This is the result of carefully applying the distributive property and combining like terms.

Mastering Polynomial Simplification: Key Takeaways

Simplifying polynomial expressions is a crucial skill in algebra. By understanding the distributive property and following a systematic approach, you can confidently tackle complex polynomial multiplications. Remember to distribute each term carefully, combine like terms accurately, and present your final answer in a clear and concise manner. With practice, you'll master the art of polynomial simplification and unlock a deeper understanding of algebraic expressions. In summary, the key takeaways from this guide are:

  • The distributive property is the foundation of polynomial multiplication.
  • Distribute each term of the first polynomial across every term of the second polynomial.
  • Identify and combine like terms to simplify the expression.
  • Write the simplified expression in a clear and organized format.

By adhering to these principles, you'll be well-equipped to simplify a wide range of polynomial expressions, paving the way for further exploration of algebraic concepts.

Practice Problems: Sharpen Your Polynomial Simplification Skills

To solidify your understanding of polynomial simplification, let's delve into some practice problems. These exercises will provide you with an opportunity to apply the concepts and techniques we've discussed, further enhancing your skills.

Practice Problem 1

Simplify the expression: (x + 2)(x - 3)

This problem involves multiplying two binomials, which is a common scenario in algebra. Apply the distributive property, carefully multiplying each term of the first binomial by each term of the second binomial. Then, combine like terms to arrive at the simplified expression.

Practice Problem 2

Simplify the expression: (2x - 1)(x² + 3x - 2)

This problem extends the previous one by involving a binomial and a trinomial. The process remains the same: distribute each term of the binomial across every term of the trinomial, and then combine like terms. Pay close attention to the signs of the terms to ensure accuracy.

Practice Problem 3

Simplify the expression: (x² + x + 1)(x² - x + 1)

This problem presents a slightly more complex scenario, involving two trinomials. However, the fundamental principles of polynomial multiplication still apply. Distribute each term of the first trinomial across every term of the second trinomial, and then carefully combine like terms. This exercise will test your ability to manage multiple terms and maintain accuracy throughout the simplification process.

By working through these practice problems, you'll gain valuable experience in applying the techniques of polynomial simplification. Remember to show your work, double-check your calculations, and take your time to ensure accuracy. With consistent practice, you'll become proficient in simplifying polynomial expressions, a skill that will serve you well in various mathematical contexts.

Conclusion: Mastering Polynomial Simplification

In conclusion, simplifying polynomial expressions is an essential skill in algebra, with applications ranging from solving equations to modeling real-world phenomena. By understanding the distributive property and employing a systematic approach, you can confidently tackle even complex polynomial multiplications. Remember to distribute each term carefully, combine like terms accurately, and present your final answer in a clear and concise manner. With practice and perseverance, you'll master the art of polynomial simplification, unlocking a deeper understanding of algebraic expressions and paving the way for further mathematical exploration.