Expanding (6r-1)(-8r-3) A Comprehensive Guide To Polynomial Multiplication

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In the realm of mathematics, understanding how to manipulate and simplify expressions is a cornerstone of problem-solving. One fundamental skill is the ability to multiply polynomials, which involves distributing terms and combining like terms to arrive at a simplified expression. This article will dissect a specific polynomial multiplication problem:

(6r - 1)(-8r - 3)

We will explore the step-by-step process of expanding this expression and arrive at the correct product. We will also discuss the underlying principles and techniques involved, ensuring a comprehensive understanding of the topic.

Expanding the Expression: A Step-by-Step Approach

To determine the product of (6r - 1) and (-8r - 3), we employ the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Let's break down the process:

  1. First: Multiply the first terms of each binomial: (6r) * (-8r) = -48r²
  2. Outer: Multiply the outer terms of the expression: (6r) * (-3) = -18r
  3. Inner: Multiply the inner terms of the expression: (-1) * (-8r) = 8r
  4. Last: Multiply the last terms of each binomial: (-1) * (-3) = 3

Now, we combine these individual products:

-48r² - 18r + 8r + 3

Next, we simplify the expression by combining like terms, which are the terms with the same variable and exponent. In this case, -18r and 8r are like terms:

-48r² + (-18r + 8r) + 3

-48r² - 10r + 3

Therefore, the product of (6r - 1) and (-8r - 3) is -48r² - 10r + 3. This result corresponds to option A in the given multiple-choice options.

Delving Deeper: The Distributive Property and Polynomial Multiplication

The distributive property is the bedrock of polynomial multiplication. It states that for any numbers a, b, and c:

a(b + c) = ab + ac

This property extends to binomials and polynomials with multiple terms. When multiplying two binomials, we are essentially distributing each term of the first binomial over the second binomial.

In our example, we can visualize the distribution as follows:

(6r - 1)(-8r - 3) = 6r(-8r - 3) - 1(-8r - 3)

Then, we apply the distributive property again:

6r(-8r) + 6r(-3) - 1(-8r) - 1(-3)

This leads us back to the same individual products we obtained using the FOIL method: -48r², -18r, 8r, and 3.

Understanding the distributive property provides a robust framework for multiplying polynomials of any size, not just binomials. For instance, to multiply a binomial by a trinomial, we would distribute each term of the binomial over all three terms of the trinomial.

The Importance of Combining Like Terms

Combining like terms is a crucial step in simplifying polynomial expressions. Like terms are terms that have the same variable raised to the same power. We can combine like terms by adding or subtracting their coefficients (the numerical part of the term).

In our example, -18r and 8r are like terms because they both have the variable r raised to the power of 1. To combine them, we simply add their coefficients: -18 + 8 = -10. This gives us the term -10r.

Failing to combine like terms will result in an unsimplified expression, which may not be the desired answer, especially in multiple-choice questions or more complex problems. It is always best practice to simplify expressions as much as possible.

Common Mistakes and How to Avoid Them

Polynomial multiplication is a straightforward process, but there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

  • Sign Errors: Pay close attention to the signs (positive or negative) of the terms. A common mistake is to forget to distribute the negative sign when multiplying a negative term. For example, in our problem, -1(-8r) should result in +8r, not -8r.
  • Incorrect Exponent Arithmetic: Remember the rules of exponents. When multiplying terms with the same base, you add the exponents. For example, r * r = r^(1+1) = r². A common mistake is to multiply the exponents instead of adding them.
  • Forgetting to Distribute: Ensure that each term in the first polynomial is multiplied by each term in the second polynomial. The FOIL method is a helpful mnemonic to remember this for binomial multiplication.
  • Failing to Combine Like Terms: As mentioned earlier, always simplify the expression by combining like terms. This ensures that you have the most concise and accurate answer.

By carefully applying the distributive property, paying attention to signs and exponents, and combining like terms, you can confidently multiply polynomials and avoid these common errors.

Practice Makes Perfect: Applying the Concepts

The best way to master polynomial multiplication is through practice. Let's consider a few more examples:

  1. (2x + 3)(x - 4)

    • First: (2x)(x) = 2x²
    • Outer: (2x)(-4) = -8x
    • Inner: (3)(x) = 3x
    • Last: (3)(-4) = -12

    Combining terms: 2x² - 8x + 3x - 12 = 2x² - 5x - 12

  2. (y - 5)(y + 2)

    • First: (y)(y) = y²
    • Outer: (y)(2) = 2y
    • Inner: (-5)(y) = -5y
    • Last: (-5)(2) = -10

    Combining terms: y² + 2y - 5y - 10 = y² - 3y - 10

  3. (3a - 1)(2a - 5)

    • First: (3a)(2a) = 6a²
    • Outer: (3a)(-5) = -15a
    • Inner: (-1)(2a) = -2a
    • Last: (-1)(-5) = 5

    Combining terms: 6a² - 15a - 2a + 5 = 6a² - 17a + 5

By working through these examples, you can solidify your understanding of polynomial multiplication and build confidence in your ability to solve similar problems.

Real-World Applications of Polynomial Multiplication

Polynomial multiplication isn't just an abstract mathematical concept; it has practical applications in various fields. Here are a few examples:

  • Geometry: Calculating the area of a rectangle or the volume of a rectangular prism often involves multiplying polynomials. For instance, if the length of a rectangle is represented by (x + 3) and the width is represented by (x - 2), the area can be found by multiplying these two binomials.
  • Physics: Polynomials are used to model projectile motion, where the height of an object is described as a function of time. Multiplying polynomials can help determine the object's position at different times.
  • Engineering: Polynomials are used in circuit analysis, signal processing, and control systems. Multiplying polynomials can help engineers design and analyze these systems.
  • Economics: Polynomials can model cost, revenue, and profit functions. Multiplying polynomials can help businesses analyze their financial performance.

These are just a few examples of how polynomial multiplication is used in the real world. Understanding this concept can open doors to various career paths and problem-solving opportunities.

Conclusion: Mastering the Art of Polynomial Multiplication

In conclusion, multiplying polynomials is a fundamental skill in mathematics with applications across various fields. By understanding the distributive property, mastering the FOIL method, and practicing regularly, you can confidently expand and simplify polynomial expressions. Remember to pay attention to signs, exponents, and like terms to avoid common errors. The product of (6r - 1) and (-8r - 3) is indeed -48r² - 10r + 3, as determined through our step-by-step analysis. Embrace the challenge of polynomial multiplication, and you'll unlock a powerful tool for problem-solving in mathematics and beyond.