Product Of -3 And Trinomial Factoring Explained

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Let's dive deep into understanding the product of -3 and the trinomial resulting from factoring out (-3) from an expression. The problem states that when you factored out (-3), you obtained (-3)(x^2) + (-3)(-7x) + (-3)(10). The question then asks us to identify the correct product that this expression simplifies to, presenting us with the equation -3x^2 + 21x - 30 = ? and three potential answers:

A. (-3)(-x^2 + 7x + (-10)) B. (-3)(x^2 - 7x + 10) C. (-3)(x^2 + 7x + 10)

To solve this, we need to meticulously examine the initial factored expression and then carefully distribute the -3 to arrive at the correct trinomial. This process involves understanding the distributive property and how it applies to both positive and negative numbers. Let's break down each step to ensure a clear comprehension of the solution.

Understanding the Distributive Property The distributive property is a fundamental concept in algebra. It allows us to multiply a single term by multiple terms within a set of parentheses. The property is expressed mathematically as:

a(b + c) = ab + ac

In simpler terms, you multiply the term outside the parentheses (a) by each term inside the parentheses (b and c) separately and then add the results. This principle extends to expressions with more than two terms inside the parentheses, such as trinomials. When dealing with negative numbers, it's crucial to pay close attention to the signs to ensure accuracy.

Analyzing the Factored Expression: (-3)(x^2) + (-3)(-7x) + (-3)(10)

Our starting point is the factored expression: (-3)(x^2) + (-3)(-7x) + (-3)(10). This expression represents the initial step of factoring out -3 from a larger polynomial. To find the correct product, we need to simplify this expression by performing the multiplications. Let's tackle each term individually:

  1. (-3)(x^2): This is a straightforward multiplication. Multiplying -3 by x^2 results in -3x^2.
  2. (-3)(-7x): Here, we are multiplying two negative terms. Remember that a negative times a negative yields a positive. So, (-3) * (-7x) = 21x.
  3. (-3)(10): This is a simple multiplication of a negative and a positive number. A negative times a positive is a negative. Thus, (-3) * (10) = -30.

Combining these results, we get the trinomial -3x^2 + 21x - 30. This is the expanded form of the factored expression, and it's the key to identifying the correct answer choice.

Matching the Trinomial with the Answer Choices

Now that we have the trinomial -3x^2 + 21x - 30, we can compare it to the answer choices provided. The original question presents the equation -3x^2 + 21x - 30 = ?, followed by three options:

A. (-3)(-x^2 + 7x + (-10)) B. (-3)(x^2 - 7x + 10) C. (-3)(x^2 + 7x + 10)

Our goal is to determine which of these options, when expanded, yields the trinomial -3x^2 + 21x - 30. Let's examine each option:

  • Option A: (-3)(-x^2 + 7x + (-10))
    • Distribute the -3:
      • (-3)(-x^2) = 3x^2
      • (-3)(7x) = -21x
      • (-3)(-10) = 30
    • The resulting trinomial is 3x^2 - 21x + 30. This does not match our target trinomial of -3x^2 + 21x - 30, so Option A is incorrect.
  • Option B: (-3)(x^2 - 7x + 10)
    • Distribute the -3:
      • (-3)(x^2) = -3x^2
      • (-3)(-7x) = 21x
      • (-3)(10) = -30
    • The resulting trinomial is -3x^2 + 21x - 30. This perfectly matches our target trinomial, making Option B the correct answer.
  • Option C: (-3)(x^2 + 7x + 10)
    • Distribute the -3:
      • (-3)(x^2) = -3x^2
      • (-3)(7x) = -21x
      • (-3)(10) = -30
    • The resulting trinomial is -3x^2 - 21x - 30. This does not match our target trinomial, so Option C is incorrect.

Therefore, after careful analysis and distribution, we've confirmed that Option B, (-3)(x^2 - 7x + 10), is the correct answer because it yields the trinomial -3x^2 + 21x - 30 when expanded. This demonstrates the importance of understanding the distributive property and paying close attention to signs when dealing with negative numbers in algebraic expressions.

To accurately find the product of -3 and the trinomial derived from factoring (-3), a systematic approach is essential. The core of the problem lies in correctly applying the distributive property and managing the signs of the terms involved. Let's outline the crucial steps to solve this type of problem:

  1. Identify the Factored Expression: Begin by clearly identifying the factored expression. In this case, it's given as (-3)(x^2) + (-3)(-7x) + (-3)(10). This expression is the result of factoring out -3 from an original trinomial, and it's the foundation for finding the product.

  2. Apply the Distributive Property: The distributive property is the cornerstone of expanding expressions like this. Remember that the distributive property states that a(b + c) = ab + ac. Apply this property to each term within the expression. This means multiplying the -3 by each term inside the parentheses or, in this case, each term of the trinomial separately.

  3. Perform the Multiplications: Execute each multiplication carefully, paying close attention to the signs. This is where precision is paramount. Here's how it breaks down for our expression:

    • (-3)(x^2) = -3x^2
    • (-3)(-7x) = 21x (Remember, a negative times a negative is a positive)
    • (-3)(10) = -30

  4. Combine the Terms: After performing the multiplications, combine the resulting terms to form the expanded trinomial. In our example, this gives us -3x^2 + 21x - 30. This trinomial is the simplified product of -3 and the terms inside the parentheses.

  5. Compare with Answer Choices: Once you have the expanded trinomial, compare it with the answer choices provided. The correct answer choice will be the one that, when expanded, matches the trinomial you've calculated. This step requires you to apply the distributive property in reverse, essentially factoring out the -3 from your calculated trinomial to see if it matches any of the given options.

  6. Verify the Solution: To ensure accuracy, it's always a good practice to verify your solution. You can do this by substituting a numerical value for x into both the original factored expression and your final trinomial. If both expressions yield the same result, your solution is likely correct. For instance, you could substitute x = 2 into both expressions and compare the outcomes.

When working with algebraic expressions, particularly those involving the distributive property and negative numbers, several common pitfalls can lead to errors. Recognizing these pitfalls and implementing strategies to avoid them is crucial for achieving accurate results. Let's explore some of these common mistakes and how to prevent them:

  1. Sign Errors: One of the most frequent errors is making mistakes with signs, especially when multiplying negative numbers. Remember the fundamental rules:

    • A negative times a negative equals a positive.
    • A negative times a positive equals a negative.

    To avoid sign errors, always double-check your multiplications and divisions, paying close attention to the signs of the numbers involved. Using parentheses can help clarify the operations and prevent confusion. For example, writing (-3) * (-7x) instead of -3 * -7x can make the operation clearer.

  2. Incorrect Distribution: Another common mistake is failing to distribute the term outside the parentheses to all terms inside. For example, in the expression -3(x^2 - 7x + 10), the -3 must be multiplied by x^2, -7x, and 10. Forgetting to multiply by one of the terms will lead to an incorrect result.

    To avoid this, use the