Multiplying Binomials A Step-by-Step Guide To (3x+4)(5x-2)

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In this comprehensive guide, we will delve into the process of multiplying binomials, specifically focusing on the expression (3x+4)(5x-2). Binomial multiplication is a fundamental concept in algebra, and mastering it is crucial for success in higher-level mathematics. We will break down the steps involved, providing clear explanations and examples along the way. By the end of this article, you will have a solid understanding of how to multiply binomials confidently and accurately.

Understanding Binomials

Before we dive into the multiplication process, let's first define what binomials are. A binomial is an algebraic expression that consists of two terms, each of which can be a constant, a variable, or a product of constants and variables. The terms are connected by either an addition or subtraction sign. For example, (3x+4) and (5x-2) are both binomials.

In the binomial (3x+4), '3x' and '4' are the two terms. '3x' is a variable term, where '3' is the coefficient and 'x' is the variable. '4' is a constant term. Similarly, in the binomial (5x-2), '5x' and '-2' are the terms, with '5x' being a variable term and '-2' being a constant term.

Understanding the structure of binomials is essential for grasping the multiplication process. When multiplying binomials, we are essentially distributing each term of the first binomial across each term of the second binomial. This process is often referred to as the FOIL method, which stands for First, Outer, Inner, Last. We will explore this method in detail in the following sections.

The FOIL Method: A Step-by-Step Approach

The FOIL method is a widely used technique for multiplying binomials. It provides a systematic way to ensure that each term of the first binomial is multiplied by each term of the second binomial. FOIL is an acronym that stands for:

  • First: Multiply the first terms of each binomial.
  • Outer: Multiply the outer terms of the binomials.
  • Inner: Multiply the inner terms of the binomials.
  • Last: Multiply the last terms of each binomial.

Let's apply the FOIL method to the expression (3x+4)(5x-2). We will go through each step in detail:

1. First Terms

The first terms of the binomials are '3x' and '5x'. Multiplying these terms gives us:

(3x) * (5x) = 15x²

Here, we multiply the coefficients (3 and 5) to get 15, and we multiply the variables (x and x) to get x². Remember that when multiplying variables with exponents, we add the exponents. In this case, x has an exponent of 1 (x¹), so x¹ * x¹ = x¹⁺¹ = x².

2. Outer Terms

The outer terms of the binomials are '3x' and '-2'. Multiplying these terms gives us:

(3x) * (-2) = -6x

Here, we multiply the coefficient 3 by -2 to get -6, and we keep the variable 'x'.

3. Inner Terms

The inner terms of the binomials are '4' and '5x'. Multiplying these terms gives us:

(4) * (5x) = 20x

Here, we multiply the constant 4 by the coefficient 5 to get 20, and we keep the variable 'x'.

4. Last Terms

The last terms of the binomials are '4' and '-2'. Multiplying these terms gives us:

(4) * (-2) = -8

Here, we multiply the constants 4 and -2 to get -8.

Combining Like Terms

After applying the FOIL method, we have the following expression:

15x² - 6x + 20x - 8

The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, '-6x' and '20x' are like terms because they both have the variable 'x' raised to the power of 1.

To combine like terms, we add their coefficients. In this case, we add -6 and 20:

-6x + 20x = 14x

Now, we can rewrite the expression with the combined like terms:

15x² + 14x - 8

This is the simplified form of the expression after multiplying the binomials (3x+4)(5x-2).

The Final Result

Therefore, the result of multiplying the binomials (3x+4)(5x-2) is:

15x² + 14x - 8

This corresponds to option C in the given choices.

Common Mistakes to Avoid

When multiplying binomials, there are several common mistakes that students often make. Being aware of these mistakes can help you avoid them and ensure accuracy in your calculations. Here are some of the most common mistakes:

1. Forgetting to Multiply All Terms

The most common mistake is forgetting to multiply all terms of the first binomial by all terms of the second binomial. This often happens when students try to take shortcuts or rush through the process. To avoid this, always use the FOIL method or a similar systematic approach to ensure that each term is multiplied correctly.

2. Incorrectly Multiplying Signs

Another common mistake is incorrectly multiplying signs. Remember that a positive number multiplied by a negative number results in a negative number, and a negative number multiplied by a negative number results in a positive number. Pay close attention to the signs of the terms when multiplying.

3. Incorrectly Combining Like Terms

Combining like terms incorrectly is another frequent error. Make sure that you are only combining terms that have the same variable raised to the same power. For example, you cannot combine 15x² and 14x because they have different powers of x.

4. Misapplying the Distributive Property

The distributive property is the foundation of binomial multiplication. Misunderstanding or misapplying it can lead to errors. Ensure you understand how each term distributes across the other binomial.

Practice Problems

To solidify your understanding of multiplying binomials, it's essential to practice. Here are a few practice problems for you to try:

  1. (2x + 3)(x - 1)
  2. (4x - 2)(3x + 5)
  3. (x + 6)(x - 6)
  4. (5x + 1)(2x - 4)

Work through these problems using the FOIL method and remember to combine like terms. You can check your answers with an online calculator or ask your teacher for assistance if needed.

Conclusion

Multiplying binomials is a crucial skill in algebra. By understanding the FOIL method and practicing regularly, you can master this skill and confidently solve a wide range of algebraic problems. Remember to take your time, pay attention to the details, and avoid common mistakes. With consistent effort, you will become proficient in multiplying binomials and excel in your mathematics studies. The key is to focus on each step, outer terms, inner terms, and last terms, ensuring a thorough application of the distributive property. Mastering this technique is a strong step toward more advanced algebraic concepts.