Simplifying Polynomials A Guide To Solving 9x²(4x + 2x² - 1)

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Polynomial expressions are fundamental in algebra, and simplifying them is a crucial skill for any mathematics student. This article will guide you through the process of simplifying the polynomial expression 9x²(4x + 2x² - 1), providing a clear, step-by-step explanation to help you master this type of problem. We will explore the underlying principles, demonstrate the distributive property, and arrive at the correct simplified form. Understanding how to simplify polynomial expressions like this one is essential for tackling more advanced algebraic concepts and problem-solving in various mathematical contexts.

Understanding Polynomial Expressions

Before we dive into the simplification process, let's first understand what a polynomial expression is. A polynomial expression is a mathematical expression consisting of variables (usually denoted by letters like x, y, or z), coefficients (numbers multiplying the variables), and non-negative integer exponents. These terms are combined using addition, subtraction, and multiplication. For example, 3x² + 2x - 1 is a polynomial expression.

In our case, the expression 9x²(4x + 2x² - 1) involves a monomial (9x²) multiplied by a trinomial (4x + 2x² - 1). To simplify this expression, we need to apply the distributive property.

The Distributive Property: The Key to Simplification

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a group of terms inside parentheses. It states that for any numbers a, b, and c:

a(b + c) = ab + ac

This property extends to expressions with more than two terms inside the parentheses. For instance:

a(b + c + d) = ab + ac + ad

In simpler terms, you multiply the term outside the parentheses by each term inside the parentheses. This is the core principle we'll use to simplify our given expression.

Step-by-Step Simplification of 9x²(4x + 2x² - 1)

Now, let's apply the distributive property to simplify the polynomial expression 9x²(4x + 2x² - 1). We will break down the process into manageable steps.

Step 1: Distribute 9x² to each term inside the parentheses

We need to multiply 9x² by each term in the trinomial (4x + 2x² - 1). This gives us:

(9x²)(4x) + (9x²)(2x²) + (9x²)(-1)

Step 2: Multiply the coefficients and add the exponents

When multiplying terms with exponents, we multiply the coefficients and add the exponents of the variables. Remember that x is the same as x¹.

  • (9x²)(4x) = 9 * 4 * x² * x¹ = 36x³
  • (9x²)(2x²) = 9 * 2 * x² * x² = 18x⁴
  • (9x²)(-1) = -9x²

So, our expression now becomes:

36x³ + 18x⁴ - 9x²

Step 3: Rearrange the terms in descending order of exponents

It is standard practice to write polynomial expressions with the terms arranged in descending order of their exponents. This makes the expression easier to read and compare with other polynomials. Therefore, we rearrange the terms as follows:

18x⁴ + 36x³ - 9x²

Therefore, the simplified form of 9x²(4x + 2x² - 1) is 18x⁴ + 36x³ - 9x². This matches option A.

Analyzing the Answer Choices

Now, let's look at the given answer choices and understand why the other options are incorrect:

  • A. 18x⁴ + 36x³ - 9x² (Correct) - As we derived in the step-by-step solution, this is the correct simplified form of the expression.
  • B. 18x⁴ - 36x³ + 9x² (Incorrect) - This option has the wrong signs for the terms 36x³ and -9x². It seems like there was an error in distributing the 9x² or in multiplying the coefficients and variables.
  • C. 36x⁴ + 18x³ - 9x² (Incorrect) - This option has the coefficients of the x⁴ and x³ terms swapped. It suggests a misunderstanding in multiplying the coefficients or in rearranging the terms.
  • D. 36x⁴ - 13x³ + 9x² (Incorrect) - This option has incorrect coefficients for all terms. This indicates a significant error in applying the distributive property and simplifying the terms.

By carefully analyzing the steps and comparing them with the answer choices, we can confidently identify the correct simplification and understand the mistakes that lead to the other incorrect options.

Common Mistakes to Avoid

Simplifying polynomial expressions involves several steps, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

  1. Incorrectly Applying the Distributive Property: Forgetting to multiply the term outside the parentheses by each term inside is a common error. Ensure you distribute the term to every term within the parentheses.
  2. Mistakes with Signs: Pay close attention to the signs (positive or negative) when multiplying. A negative sign multiplied by a positive sign results in a negative sign, and a negative sign multiplied by a negative sign results in a positive sign.
  3. Adding Exponents Incorrectly: Remember to add the exponents when multiplying terms with the same base (e.g., x² * x = x³). A common mistake is to multiply the exponents instead of adding them.
  4. Combining Unlike Terms: Only combine terms that have the same variable and exponent. For example, you can combine 3x² and 5x², but you cannot combine 3x² and 5x.
  5. Forgetting to Rearrange Terms: While not strictly an error, leaving the terms in the wrong order (not in descending order of exponents) can make the expression appear less organized and harder to compare with other polynomials.

By being aware of these common mistakes, you can improve your accuracy and simplify polynomial expressions with greater confidence.

Practice Problems

To solidify your understanding of simplifying polynomial expressions, try solving these practice problems:

  1. Simplify: 5x(2x² - 3x + 1)
  2. Simplify: -3y²(4y³ + 2y - 5)
  3. Simplify: 2a³(a² - 5a + 3)

By working through these problems, you can reinforce the steps and techniques discussed in this article and develop your skills in simplifying polynomial expressions.

Conclusion

Simplifying polynomial expressions is a fundamental skill in algebra. By understanding the distributive property and following a step-by-step approach, you can confidently simplify expressions like 9x²(4x + 2x² - 1). Remember to distribute carefully, pay attention to signs and exponents, and rearrange the terms in descending order. By avoiding common mistakes and practicing regularly, you can master this important skill and excel in your mathematics studies. The correct simplification of 9x²(4x + 2x² - 1) is indeed 18x⁴ + 36x³ - 9x², option A.