Simplifying Polynomial Expressions A Step By Step Guide

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Hey guys! Let's dive into a math problem that might seem a bit intimidating at first glance, but trust me, we'll break it down together. We're going to tackle the simplification of the expression (5x - 6)(3x² - 4x - 3). It's a classic algebra problem that tests our understanding of polynomial multiplication and combining like terms. Stick with me, and you'll master this in no time!

Breaking Down the Problem

So, when we look at (5x - 6)(3x² - 4x - 3), what we're really being asked to do is multiply these two expressions together. The first expression, (5x - 6), is a binomial because it has two terms. The second expression, (3x² - 4x - 3), is a trinomial because it has three terms. To multiply these, we're going to use a method that ensures we multiply each term in the first expression by each term in the second expression. Some people call this the distributive property, or the FOIL method (First, Outer, Inner, Last) when dealing with two binomials. But since we have a binomial and a trinomial here, we'll focus on the distributive property to make sure we get every term.

To kick things off, we'll take the first term in the binomial, which is 5x, and multiply it by each term in the trinomial. That means we'll multiply 5x by 3x², then by -4x, and finally by -3. After that, we'll do the same thing with the second term in the binomial, which is -6. We'll multiply -6 by 3x², then by -4x, and finally by -3. It might seem like a lot of steps, but it's really just about being organized and making sure we don't miss any multiplications. Once we've done all the multiplications, we'll have a long expression with several terms. The next step is to simplify by combining any like terms. Like terms are terms that have the same variable raised to the same power. For example, 3x² and -2x² are like terms because they both have x². We can add or subtract the coefficients (the numbers in front of the variables) of like terms to simplify the expression. This is where the problem really comes together, and we see how all the individual multiplications combine to form the final answer.

Step-by-Step Multiplication

Let's get into the nitty-gritty and walk through the multiplication step by step. Remember, we're starting with (5x - 6)(3x² - 4x - 3). First, we distribute the 5x across the trinomial:

  • 5x * 3x² = 15x³ (Remember, when multiplying variables with exponents, we add the exponents. So, x * x² = x^(1+2) = x³)
  • 5x * -4x = -20x²
  • 5x * -3 = -15x

Now, we distribute the -6 across the trinomial:

  • -6 * 3x² = -18x²
  • -6 * -4x = 24x (A negative times a negative is a positive!)
  • -6 * -3 = 18

So, after distributing, our expression looks like this: 15x³ - 20x² - 15x - 18x² + 24x + 18. It's a bit of a jumble right now, but don't worry, we're about to clean it up!

Combining Like Terms

The next step, and a crucial one, is combining like terms. This is where we gather the terms that have the same variable and exponent and add or subtract their coefficients. It's like sorting a pile of coins – we group the pennies together, the nickels together, and so on. In our expression, we have terms with x³, x², x, and constant terms (numbers without variables).

Let's identify the like terms in our expression: 15x³ - 20x² - 15x - 18x² + 24x + 18

  • We have one x³ term: 15x³
  • We have two x² terms: -20x² and -18x²
  • We have two x terms: -15x and 24x
  • We have one constant term: 18

Now, let's combine them:

  • 15x³ stays as is since there are no other x³ terms.
  • -20x² - 18x² = -38x² (We add the coefficients: -20 + (-18) = -38)
  • -15x + 24x = 9x (We add the coefficients: -15 + 24 = 9)
  • 18 stays as is since there are no other constant terms.

Putting it all together, we get: 15x³ - 38x² + 9x + 18. And there you have it! We've successfully simplified the expression.

Identifying the Correct Option

Okay, so we've done the hard work of simplifying the expression. Now, let's circle back to the original question and see which of the answer choices matches our simplified expression. We found that (5x - 6)(3x² - 4x - 3) simplifies to 15x³ - 38x² + 9x + 18.

Looking at the answer choices, we have:

A. 15x³ - 38x² + 9x - 18 B. 15x³ + 38x² - 9x + 18 C. 15x³ - 38x² + 9x + 18 D. 15x³ + 38x² - 9x - 18

Comparing our simplified expression to the choices, we can clearly see that option C, 15x³ - 38x² + 9x + 18, is the correct match. Options A, B, and D have different signs for some of the terms, which means they are not equivalent to our simplified expression. So, the correct answer is C!

Common Mistakes to Avoid

Alright, let's talk about some common pitfalls that students often encounter when tackling problems like this. Recognizing these mistakes can help you avoid them and boost your confidence in algebra.

One of the biggest mistakes is forgetting to distribute properly. Remember, each term in the first expression needs to be multiplied by each term in the second expression. It's easy to miss a multiplication, especially when there are multiple terms involved. A good way to prevent this is to be methodical and write out each multiplication step by step, like we did earlier. Double-checking your work can also help catch any missed multiplications.

Another common mistake is incorrectly multiplying the coefficients or exponents. When multiplying terms with exponents, remember to add the exponents, not multiply them. For example, x² * x³ = x^(2+3) = x⁵, not x⁶. Similarly, be careful when multiplying the coefficients. A simple arithmetic error can throw off the entire solution. Again, double-checking your work can help catch these errors.

Sign errors are also a frequent source of mistakes. Pay close attention to the signs of the terms, especially when distributing a negative number. Remember that a negative times a negative is a positive, and a negative times a positive is a negative. Writing out the steps clearly and carefully can help you keep track of the signs.

Finally, combining unlike terms is a mistake that can easily happen if you're not careful. Remember, you can only combine terms that have the same variable raised to the same power. For example, you can combine 3x² and -2x², but you cannot combine 3x² and 2x. Make sure you're only adding or subtracting the coefficients of like terms.

By being aware of these common mistakes and taking steps to avoid them, you'll be well on your way to mastering polynomial multiplication and simplification!

Why This Matters

Now, you might be wondering,