Multiply And Simplify (v+1)(3v-3): A Step-by-Step Guide

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Hey guys! Today, we're diving into a common algebra problem: multiplying and simplifying expressions. Specifically, we're going to tackle the expression (v+1)(3v-3). This is a classic example that involves the distributive property and combining like terms. Don't worry, it's not as intimidating as it sounds! We'll break it down step-by-step so you can follow along easily. Mastering this type of problem is crucial for success in algebra and beyond, as it forms the foundation for more complex mathematical concepts. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have two binomials, (v+1) and (3v-3), and we need to multiply them together. This means every term in the first binomial must be multiplied by every term in the second binomial. Then, we need to simplify the resulting expression by combining any like terms. Remember, like terms are terms that have the same variable raised to the same power. For example, 3v and -3v are like terms, but 3v and 3vĀ² are not. This process is often referred to as expanding and then simplifying the expression. It's a fundamental skill in algebra, so let's make sure we nail it! Understanding the underlying principles here will help you tackle a wide range of similar problems with confidence.

Method 1: The Distributive Property (FOIL Method)

One of the most common ways to multiply binomials is using the distributive property, often remembered by the acronym FOIL:

  • 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 this to our expression, (v+1)(3v-3):

  1. First: v * 3v = 3vĀ²
  2. Outer: v * -3 = -3v
  3. Inner: 1 * 3v = 3v
  4. Last: 1 * -3 = -3

Now, we add all these terms together: 3vĀ² - 3v + 3v - 3. Notice that we have a -3v and a +3v. These are like terms, and when we combine them, they cancel each other out. This is a common occurrence when simplifying expressions, so keep an eye out for opportunities to simplify! The FOIL method is a fantastic tool for multiplying binomials, and with a little practice, you'll be able to apply it quickly and accurately.

Step-by-Step Solution using FOIL

Let's break down the process of multiplying (v+1)(3v-3) using the FOIL method in more detail:

  1. Multiply the First terms:

    • We start by multiplying the first terms of each binomial: v from (v+1) and 3v from (3v-3).
    • This gives us v * 3v = 3vĀ². Remember, when multiplying variables, we add their exponents. In this case, v has an exponent of 1, so vĀ¹ * vĀ¹ = vĀ¹āŗĀ¹ = vĀ².
    • This first step is crucial as it sets the stage for the rest of the calculation. Getting the initial multiplication correct is paramount to arriving at the correct final answer.

  2. Multiply the Outer terms:

    • Next, we multiply the outer terms: v from (v+1) and -3 from (3v-3).
    • This gives us v * -3 = -3v. Pay close attention to the signs! A negative times a positive is a negative.
    • Keeping track of the signs is a common area where mistakes can happen, so it's always a good idea to double-check your work.

  3. Multiply the Inner terms:

    • Now, we multiply the inner terms: 1 from (v+1) and 3v from (3v-3).
    • This gives us 1 * 3v = 3v. This step is usually straightforward, but it's still important to perform it carefully.
    • Even simple multiplications need attention to detail to avoid errors.

  4. Multiply the Last terms:

    • Finally, we multiply the last terms of each binomial: 1 from (v+1) and -3 from (3v-3).
    • This gives us 1 * -3 = -3. Again, pay attention to the negative sign.
    • This final multiplication completes the expansion of the binomials.

  5. Combine the terms:

    • Now, we add all the terms we've calculated together: 3vĀ² - 3v + 3v - 3.
    • Notice that we have two terms with 'v': -3v and +3v. These are like terms, meaning they have the same variable raised to the same power.
    • Like terms can be combined by adding or subtracting their coefficients (the numbers in front of the variable).

  6. Simplify the expression:

    • In this case, -3v + 3v = 0. They cancel each other out.
    • So, our simplified expression is 3vĀ² - 3.
    • This is the final answer! We've successfully multiplied and simplified the expression (v+1)(3v-3).

Method 2: The Distributive Property (Expanded)

Another way to approach this problem is to use the distributive property in a more expanded form. Instead of using FOIL as a shortcut, we'll explicitly distribute each term of the first binomial across the second binomial. This method can be particularly helpful when dealing with larger expressions or when you want to ensure you don't miss any terms. Let's see how it works with (v+1)(3v-3):

  1. Distribute the 'v' from the first binomial across the second binomial:

    • v * (3v - 3) = v * 3v - v * 3 = 3vĀ² - 3v
    • We've taken the first term of the first binomial and multiplied it by both terms of the second binomial.
    • This is the first part of the expansion process.

  2. Distribute the '1' from the first binomial across the second binomial:

    • 1 * (3v - 3) = 1 * 3v - 1 * 3 = 3v - 3
    • Now we've taken the second term of the first binomial and multiplied it by both terms of the second binomial.
    • This completes the distribution process.

  3. Combine the results:

    • Now we add the results from steps 1 and 2 together: (3vĀ² - 3v) + (3v - 3) = 3vĀ² - 3v + 3v - 3
    • This gives us the same expression we obtained using the FOIL method.
    • At this point, we're ready to simplify the expression by combining like terms.

  4. Simplify the expression:

    • As before, we combine the like terms: -3v + 3v = 0.
    • So, our simplified expression is 3vĀ² - 3. This matches the result we got using the FOIL method, which is a good sign!
    • This consistency across different methods helps confirm that our answer is correct.

Step-by-Step Solution using Expanded Distributive Property

Let's go through the expanded distributive property method with a bit more detail to ensure we're clear on each step:

  1. Distribute 'v' across (3v - 3):

    • We start by multiplying 'v' by '3v': v * 3v = 3vĀ²
    • Then, we multiply 'v' by '-3': v * -3 = -3v
    • So, distributing 'v' across (3v - 3) gives us 3vĀ² - 3v.
    • Remember, this is just one part of the full expansion.

  2. Distribute '1' across (3v - 3):

    • Next, we multiply '1' by '3v': 1 * 3v = 3v
    • Then, we multiply '1' by '-3': 1 * -3 = -3
    • So, distributing '1' across (3v - 3) gives us 3v - 3.
    • Now we have the result of distributing both terms from the first binomial.

  3. Combine the distributed terms:

    • We add the results from the previous two steps: (3vĀ² - 3v) + (3v - 3)
    • This is the crucial step where we bring all the terms together to form a single expression.
    • It's important to write down each term carefully to avoid mistakes.

  4. Write out the combined expression:

    • Combining the terms gives us: 3vĀ² - 3v + 3v - 3
    • This expression contains like terms that we can simplify.
    • Identifying and combining like terms is the key to simplifying algebraic expressions.

  5. Identify and combine like terms:

    • We have -3v and +3v, which are like terms because they both have the variable 'v' raised to the power of 1.
    • These terms can be combined by adding their coefficients: -3 + 3 = 0
    • So, -3v + 3v = 0, and these terms cancel each other out.

  6. Write the simplified expression:

    • After combining like terms, we're left with 3vĀ² - 3.
    • This is the fully simplified result of multiplying and simplifying the original expression.
    • We've successfully used the expanded distributive property to solve the problem.

Final Answer

Both methods, FOIL and the expanded distributive property, lead us to the same simplified expression: 3vĀ² - 3. So, (v+1)(3v-3) multiplied and simplified is 3vĀ² - 3. Remember, practice makes perfect! The more you work through these types of problems, the more comfortable you'll become with the process. Don't be afraid to try different approaches and see what works best for you. Algebra is a building block for more advanced math, so mastering these fundamentals is essential. Keep up the great work, and you'll be multiplying and simplifying like a pro in no time! Remember, the key is to break down the problem into smaller, manageable steps and to pay close attention to the details, especially the signs. With consistent practice, you'll find these types of problems becoming second nature. And that's a fantastic feeling when you conquer a challenging math problem!