Product Of (3+x) And (2x-5) In Standard Form Explained

by Viktoria Ivanova 55 views

Hey there, math enthusiasts! Ever found yourself staring at an algebraic expression, feeling like you're trying to decipher an ancient code? Well, you're not alone! Today, we're diving deep into the world of polynomial multiplication and standard form, using the expression (3+x)(2x-5) as our trusty guide. We'll break down each step, making sure you not only get the answer but also understand why it's the answer. Let's get started, guys!

The Initial Problem: Multiplying Binomials

So, the question is: When written in standard form, what's the product of (3+x) and (2x-5)? We've got four options:

A. 3x - 2 B. 2x² + x - 15 C. 2x² - 11x - 15 D. 6x - 15 + 2x² - 5x

At first glance, it might seem a bit daunting, especially if you're not super comfortable with multiplying binomials. But fear not! We're going to tackle this step-by-step, making it super clear and easy to follow. The core concept here is the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This nifty little trick helps us ensure we multiply each term in the first binomial by each term in the second.

Breaking Down FOIL: A Closer Look

Let's quickly recap what FOIL stands for:

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

Now, let’s apply this to our problem. Imagine we're detectives, carefully collecting each piece of evidence (each multiplication) to solve the case (find the correct answer). The FOIL method is our magnifying glass, helping us see each detail clearly.

Applying FOIL to (3+x)(2x-5)

Okay, let's roll up our sleeves and get our hands dirty with the math. We're going to systematically apply the FOIL method to (3+x)(2x-5). Think of it like following a recipe – each step is crucial for the perfect final product.

Step 1: First Terms

Multiply the first terms in each binomial: 3 * 2x. This is pretty straightforward. 3 multiplied by 2x gives us 6x. So far, so good! We've got our first piece of the puzzle. It's like finding the first clue in a mystery novel – exciting!

Step 2: Outer Terms

Next up, we multiply the outer terms: 3 * -5. Remember to pay attention to the signs! 3 multiplied by -5 equals -15. This is a crucial step, as a simple sign error can throw off the entire calculation. Imagine baking a cake and forgetting the sugar – it just wouldn't taste right!

Step 3: Inner Terms

Now, let's tackle the inner terms: x * 2x. Here, we're multiplying a variable by a term containing the same variable. Remember the rule: x times x is x². So, x * 2x equals 2x². We're building up our expression piece by piece, like constructing a Lego masterpiece.

Step 4: Last Terms

Finally, we multiply the last terms: x * -5. Again, watch those signs! x multiplied by -5 equals -5x. We've now completed all the individual multiplications. It’s like finishing all the tasks on your to-do list – a great feeling!

Combining the Terms: Putting the Puzzle Together

We've done all the hard work of multiplying using FOIL. Now comes the satisfying part: combining the terms we've found. We have:

  • 6x (from the