Polynomial Product Calculation (2x - 4) • (3x + 1) With Solution

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Hey guys! In this article, we're going to break down how to calculate the product of the polynomials (2x - 4) and (3x + 1). This is a common type of problem in algebra, and understanding how to solve it is super important. We’ll go step by step, making sure everything is clear so you can tackle similar problems with confidence. So, let’s dive in and find the correct answer together!

Understanding Polynomial Multiplication

Before we jump into the specific problem, let's quickly recap what polynomial multiplication means. When we multiply two polynomials, we're essentially applying the distributive property multiple times. This means each term in the first polynomial needs to be multiplied by each term in the second polynomial. Think of it like this: every term gets a chance to "meet" and interact with every other term.

The distributive property, in its basic form, says that a(b + c) = ab + ac. But when we have polynomials with multiple terms, we extend this idea. For instance, when multiplying (A + B) by (C + D), we get:

(A + B) • (C + D) = A • C + A • D + B • C + B • D

This might seem a bit abstract right now, but don't worry! We'll apply this exact method to our specific problem, and you'll see how it works in practice. Remembering this pattern is key to successfully multiplying any two polynomials. It’s like having a roadmap – you know exactly which steps to take to reach your destination. So, with this roadmap in mind, let’s move on to our actual calculation and see how this plays out.

Step-by-Step Calculation of (2x - 4) • (3x + 1)

Now, let's get to the heart of the matter and calculate the product of (2x - 4) and (3x + 1). We’ll take it one step at a time to ensure everything is crystal clear.

Following the distributive property method we just discussed, we need to multiply each term in the first polynomial (2x - 4) by each term in the second polynomial (3x + 1). This gives us four multiplications to perform:

  1. 2x multiplied by 3x
  2. 2x multiplied by 1
  3. -4 multiplied by 3x
  4. -4 multiplied by 1

Let’s break each of these down:

1. Multiplying 2x by 3x

When we multiply 2x by 3x, we multiply the coefficients (the numbers in front of the x) and add the exponents of x. Remember, x is the same as x^1, so:

(2x) • (3x) = 2 • 3 • x^1 • x^1 = 6x^(1+1) = 6x²

So, the first term of our product is 6x². This is a crucial step, and it sets the stage for the rest of the calculation.

2. Multiplying 2x by 1

This one's straightforward. Any term multiplied by 1 is just itself:

(2x) • (1) = 2x

So, the second term is simply 2x. Easy peasy!

3. Multiplying -4 by 3x

Here, we multiply the numbers -4 and 3, and then include the x:

(-4) • (3x) = -4 • 3 • x = -12x

This gives us -12x as our third term. Remember to keep track of the negative sign!

4. Multiplying -4 by 1

Again, multiplying by 1 is straightforward:

(-4) • (1) = -4

So, the final term is -4. We’re almost there!

Now that we've done all the individual multiplications, let's put it all together and see what we've got.

Combining the Terms

Okay, we've done the individual multiplications, and now it's time to combine all the terms we found. Remember, we had these four results:

  • 6x²
  • 2x
  • -12x
  • -4

So, we add them all together to get the product of our polynomials:

6x² + 2x + (-12x) + (-4)

Now, we need to simplify this expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, 2x and -12x are like terms because they both have x to the power of 1. We can combine them by adding their coefficients:

2x + (-12x) = 2x - 12x = -10x

So, when we combine these like terms, we get -10x. Now, let’s put everything back together. Our expression becomes:

6x² - 10x - 4

And there you have it! This is the product of the polynomials (2x - 4) and (3x + 1). We've gone through each step meticulously, so you can see exactly how we arrived at this answer. Now, let's match this result with the options given in the question.

Identifying the Correct Option

Alright, we've calculated the product of the polynomials and found it to be 6x² - 10x - 4. Now, let’s take a look at the options provided in the question to see which one matches our result:

The options were:

a) 6x² - 10x - 4 b) 6x² - 10x + 4 c) 6x² + 10x - 4 d) 6x² + 10x + 4

Comparing our result, 6x² - 10x - 4, with the options, it's clear that option a) is the correct one. Options b), c), and d) have different signs in either the middle term or the constant term, so they are not the correct products.

Therefore, the correct answer is:

a) 6x² - 10x - 4

We've successfully navigated through the entire problem, from understanding the principles of polynomial multiplication to performing the calculations and identifying the correct answer. Pat yourself on the back – you’ve done great!

Final Answer

So, the final answer to the question is:

(a) 6x² - 10x - 4

We started by understanding the basic principles of polynomial multiplication, then we broke down the specific problem into manageable steps, and finally, we combined the results to find the correct answer. This step-by-step approach is key to solving many algebraic problems. By taking your time and working through each part methodically, you can avoid mistakes and build your confidence. Keep practicing, and you'll become a pro at polynomial multiplication in no time!

In this article, we walked through a typical polynomial multiplication problem. Remember, the key is to distribute each term correctly and then combine like terms. With practice, these types of problems become much easier. Keep up the great work, and you'll be mastering algebra in no time!