Polynomial Multiplication: Step-by-Step Solutions

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Hey guys! Let's dive into the world of polynomial multiplication. It might sound intimidating, but trust me, it’s like putting together puzzle pieces. We'll break down each problem step-by-step, making sure you understand the hows and whys. So grab your pencil and paper, and let’s get started!

Expanding Polynomial Expressions

In this section, we're going to tackle several polynomial multiplication problems. We'll focus on using the distributive property (or the FOIL method for binomials) to expand these expressions. Remember, the key is to multiply each term in the first polynomial by each term in the second polynomial and then simplify by combining like terms. Let’s get into the nitty-gritty!

Problem A: (2x + 3)(x - 5)

Okay, let's kick things off with our first problem: (2x + 3)(x - 5). This is a classic example of multiplying two binomials. We'll use the distributive property, which some people like to remember as the FOIL method (First, Outer, Inner, Last). This means we'll multiply the first terms, then the outer terms, then the inner terms, and finally the last terms.

  • First: Multiply the first terms in each binomial: 2x * x = 2x²
  • Outer: Multiply the outer terms: 2x * -5 = -10x
  • Inner: Multiply the inner terms: 3 * x = 3x
  • Last: Multiply the last terms: 3 * -5 = -15

Now, let's put it all together: 2x² - 10x + 3x - 15. Don't forget the crucial step: combining like terms. We have -10x and 3x, which combine to -7x. So, our final answer is:

2x² - 7x - 15

See? Not so scary when you break it down. The distributive property is your best friend here, ensuring every term gets its turn in the multiplication game.

Problem B: (4x - 5y)(2x + 3y)

Next up, we've got (4x - 5y)(2x + 3y). This one introduces another variable, y, but the same principles apply. We'll use the FOIL method again to make sure we hit all the terms.

  • First: 4x * 2x = 8x²
  • Outer: 4x * 3y = 12xy
  • Inner: -5y * 2x = -10xy
  • Last: -5y * 3y = -15y²

Combine these terms: 8x² + 12xy - 10xy - 15y². Now, let's simplify by combining like terms. We've got 12xy and -10xy, which combine to 2xy. So, our simplified expression is:

8x² + 2xy - 15y²

It’s all about being methodical and keeping track of those signs. A little bit of care goes a long way in getting the right answer.

Problem C: (-x + 7)(3x + 2)

Let's switch gears slightly with (-x + 7)(3x + 2). Notice the negative sign in front of the x term. This is a common spot for errors, so let's pay extra attention. We're still using the distributive property (or FOIL), so no changes there.

  • First: -x * 3x = -3x²
  • Outer: -x * 2 = -2x
  • Inner: 7 * 3x = 21x
  • Last: 7 * 2 = 14

Now, let’s put it all together: -3x² - 2x + 21x + 14. Combining the like terms -2x and 21x gives us 19x. Therefore, the simplified expression is:

-3x² + 19x + 14

Remember, the order of terms doesn’t technically matter, but it’s common practice to write the terms in descending order of their exponents. It just looks cleaner!

Problem D: (x - 3y)(2x + 5y)

Moving on, we have (x - 3y)(2x + 5y). This problem is similar to Problem B, with two variables. Let’s break it down using our trusty FOIL method.

  • First: x * 2x = 2x²
  • Outer: x * 5y = 5xy
  • Inner: -3y * 2x = -6xy
  • Last: -3y * 5y = -15y²

Combine the terms: 2x² + 5xy - 6xy - 15y². Now, let’s simplify. The 5xy and -6xy combine to -xy. So, the simplified expression is:

2x² - xy - 15y²

See how consistent this method is? Once you get the hang of it, these problems become almost automatic.

Problem E: (2x - y)(3x² + 2y)

Now for something a little different: (2x - y)(3x² + 2y). Here, we're multiplying a binomial by what looks like another binomial, but the second expression has an x² term. No problem! The distributive property still applies. We just need to be a little more careful with our multiplications.

  • 2x * 3x² = 6x³
  • 2x * 2y = 4xy
  • -y * 3x² = -3x²y
  • -y * 2y = -2y²

Putting it together, we get: 6x³ + 4xy - 3x²y - 2y². Are there any like terms to combine? Nope! We have x³, xy, x²y, and y² terms, all distinct. So, this is our final answer:

6x³ + 4xy - 3x²y - 2y²

Sometimes, the final expression looks a bit more complex, and that's okay. Just make sure you've multiplied everything correctly and haven't missed any combinations.

Dealing with Square Roots

Now, let's amp things up a bit by introducing square roots into our expressions. Don't worry; the fundamental principles of polynomial multiplication remain the same. We'll still be using the distributive property (FOIL), but we'll need to remember the rules for multiplying radicals.

Problem F: (√3x - 1)(2√3x + 4)

Okay, let’s tackle (√3x - 1)(2√3x + 4). This one might look intimidating with those square roots, but we’ll take it step by step. Remember, when multiplying radicals, we multiply the numbers inside the square roots and the numbers outside the square roots separately.

  • First: √3x * 2√3x = 2 * (√3 * √3) * (x * x) = 2 * 3 * x = 6x
  • Outer: √3x * 4 = 4√3x
  • Inner: -1 * 2√3x = -2√3x
  • Last: -1 * 4 = -4

Combine these: 6x + 4√3x - 2√3x - 4. Now, let's look for like terms. We have 4√3x and -2√3x, which are like terms because they both contain the same radical part (√3x). Combining them, we get 2√3x. So, our simplified expression is:

6x + 2√3x - 4

The key here is to remember that you can only combine terms that have the same radical part. If you keep that in mind, you'll be golden!

Problem G: (√5x - y)(x - √5y)

Next, we have (√5x - y)(x - √5y). This one mixes square roots with our good old friend, the variable y. Let's apply the distributive property (FOIL) as usual.

  • First: √5x * x = x√5x
  • Outer: √5x * -√5y = -√5 * √5 * √(xy) = -5√(xy)
  • Inner: -y * x = -xy
  • Last: -y * -√5y = y√5y

Putting it all together, we have: x√5x - 5√(xy) - xy + y√5y. Take a look at the terms. Are there any like terms we can combine? Nope! Each term has a different combination of variables and radicals. So, this is our final answer:

x√5x - 5√(xy) - xy + y√5y

Sometimes, you end up with an expression that looks a bit messy, and that’s perfectly fine. The important thing is that you’ve applied the multiplication rules correctly.

Problem H: (√5x - √3y)(√5x + 2√3y)

Now, let's tackle (√5x - √3y)(√5x + 2√3y). This one has square roots in both terms of both binomials, so it's a good exercise in keeping track of our multiplications. Let's dive in using FOIL.

  • First: √5x * √5x = 5x
  • Outer: √5x * 2√3y = 2√(5 * 3) * √(xy) = 2√15√(xy)
  • Inner: -√3y * √5x = -√(3 * 5) * √(xy) = -√15√(xy)
  • Last: -√3y * 2√3y = -2 * (√3 * √3) * y = -2 * 3 * y = -6y

Combine the terms: 5x + 2√15√(xy) - √15√(xy) - 6y. Notice that we have 2√15√(xy) and -√15√(xy), which are like terms. Combining them gives us √15√(xy). So, the simplified expression is:

5x + √15√(xy) - 6y

It’s all about careful multiplication and recognizing those like terms. Once you’ve got that down, you’re in great shape.

Problem I: (√3x - √2)(√3x + √2)

Last but not least in this section, we have (√3x - √2)(√3x + √2). This one is special! It’s in the form of (a - b)(a + b), which is a difference of squares. This means we can use a shortcut: (a - b)(a + b) = a² - b². Let’s see how it works.

Here, a = √3x and b = √2. So, we have:

(√3x)² - (√2)²

Squaring a square root just gives us the number inside the square root. So:

3x - 2

That's it! No need for the full FOIL process. Recognizing these special forms can save you a lot of time and effort.

3x - 2

More Multiplication Practice

Okay, let's move on to some additional multiplication problems. These might look a little different, but the same principles apply. We're still going to use the distributive property to make sure we multiply each term correctly.

Problem 10a: (-3x) - (+)

Okay, this one looks a bit incomplete as written, so let’s clarify. It seems like we’re supposed to be multiplying -3x by something, but the second part is missing. Without more information, we can’t complete this problem. However, let’s assume for a moment that there was a term intended to be multiplied by -3x. For example, if the problem was:

-3x * (2x + 1)

Then we would solve it like this:

  • -3x * 2x = -6x²
  • -3x * 1 = -3x

So, the answer would be -6x² - 3x. But without knowing the full problem, we can’t give a definitive answer.

Problem 10b: (-7xy)

Similarly, (-7xy) by itself isn’t a multiplication problem. It’s just a term. If we were supposed to multiply it by something, we’d need more information. Let's imagine we were supposed to multiply it by (3x - 2y):

-7xy * (3x - 2y)

Then we would do:

  • -7xy * 3x = -21x²y
  • -7xy * -2y = 14xy²

So, the result would be -21x²y + 14xy².

Final Thoughts

And there you have it, guys! We’ve tackled a bunch of polynomial multiplication problems, from simple binomial multiplication to expressions with square roots. Remember, the key is to take it step by step, use the distributive property (or FOIL), and carefully combine those like terms. Keep practicing, and you'll become a multiplication master in no time! Happy math-ing!