Multiply & Combine Like Terms: (2m + 7n)(m + 7n)

by Andrew McMorgan 49 views

Hey guys! Today, we're diving into some algebra to tackle the expression (2m + 7n)(m + 7n). This involves multiplying binomials and then combining any like terms to simplify the result. Sounds like fun, right? Let's break it down step by step so you can totally nail it.

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have two binomials, (2m + 7n) and (m + 7n), and we need to multiply them together. This means every term in the first binomial needs to be multiplied by every term in the second binomial. After we do that, we'll look for terms that have the same variables raised to the same powers (like terms) so we can combine them. This process is super important in algebra, and it's something you'll use all the time, so let's get comfortable with it.

Breaking Down the Binomials

So, what exactly is a binomial? A binomial is simply an algebraic expression with two terms. In our case, 2m + 7n is a binomial because it has two terms: 2m and 7n. Similarly, m + 7n is also a binomial with the terms m and 7n. The goal here is to multiply these two binomials together, making sure we account for each term properly. We're essentially distributing each term of the first binomial across each term of the second binomial. Think of it like this: each term wants to say "hi" to every other term through multiplication!

Why Combining Like Terms Matters

After we multiply everything out, we often end up with a bunch of terms. But not all terms are created equal. Like terms are those that have the same variable raised to the same power. For example, 3m and 5m are like terms because they both have the variable m raised to the power of 1. On the other hand, 3m and 5m² are not like terms because the powers of m are different. Combining like terms simplifies our expression by adding or subtracting their coefficients (the numbers in front of the variables). This makes the expression cleaner and easier to work with. It's like tidying up a messy room – you want to group similar items together to make everything more organized!

Step-by-Step Solution

Alright, let's get our hands dirty and solve this thing! We'll use the FOIL method (First, Outer, Inner, Last) to make sure we multiply each term correctly.

Applying the FOIL Method

The FOIL method is a handy way to remember how to multiply two binomials. It stands for:

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

Let's apply this to our expression, (2m + 7n)(m + 7n):

  • First: 2m * m = 2m²
  • Outer: 2m * 7n = 14mn
  • Inner: 7n * m = 7mn
  • Last: 7n * 7n = 49n²

So, after applying FOIL, we have:

2m² + 14mn + 7mn + 49n²

Combining Like Terms

Now, let's look for like terms. In our expression, we have 14mn and 7mn, which are like terms because they both have the variables m and n raised to the power of 1. We can combine these by adding their coefficients:

14mn + 7mn = 21mn

So, our expression now becomes:

2m² + 21mn + 49n²

The Final Result

We've multiplied the binomials and combined all the like terms. There are no more like terms to combine, so we're done! The simplified expression is:

2m² + 21mn + 49n²

That's it! We've successfully multiplied (2m + 7n)(m + 7n) and simplified the result. Give yourself a pat on the back – you're doing great!

Common Mistakes to Avoid

When you're multiplying and combining like terms, it's easy to make a few common mistakes. Here are some things to watch out for:

Forgetting to Distribute Properly

One of the biggest mistakes is not multiplying each term in the first binomial by each term in the second binomial. Make sure you use the FOIL method (or any other method you prefer) to keep track of all the multiplications. It's easy to accidentally skip a term, especially when you're working quickly.

Mixing Up the Signs

Be careful with your signs! A negative sign can easily get lost or misapplied, which can totally change your answer. Remember the rules for multiplying with negative numbers: a negative times a positive is negative, and a negative times a negative is positive. Double-check your signs at each step to avoid these errors.

Combining Unlike Terms

Only combine terms that are truly alike. Remember, like terms have the same variables raised to the same powers. For example, you can combine 3x² and 5x², but you can't combine 3x² and 5x. Mixing these up will lead to an incorrect simplification.

Not Simplifying Completely

Make sure you've combined all possible like terms. Sometimes, you might overlook a pair of like terms, leaving your expression not fully simplified. Double-check your work to ensure you've caught all the like terms and combined them correctly.

Practice Problems

Want to really nail this skill? Here are a few practice problems you can try. Work through them step by step, and don't forget to check your answers!

  1. (3x + 2y)(x + 4y)
  2. (a - 5b)(2a + b)
  3. (4p - 3q)(p - 2q)

Work these out using the steps we covered. Multiply, combine, and simplify! Remember the FOIL method and watch out for those common mistakes!

Conclusion

Multiplying and combining like terms is a fundamental skill in algebra. By understanding the FOIL method and being careful to combine only like terms, you can simplify complex expressions with ease. Remember to avoid common mistakes like forgetting to distribute, mixing up signs, and combining unlike terms. With a little practice, you'll become a pro at this in no time!

Keep practicing, and you'll be able to tackle any algebraic expression that comes your way. You got this! Now go out there and conquer those binomials!