Closure Property Of Multiplication Polynomials Explained

Hey guys! Let's dive into a fun math problem today that touches on the closure property of multiplication, especially when we're dealing with polynomials. Polynomials might sound intimidating, but they're really just expressions with variables and coefficients, like the ones we're about to work with. So, let's get started and make sure we understand what's going on!

What is the Closure Property?

Before we jump into the specific problem, let's quickly recap what the closure property actually means. In simple terms, a set is "closed" under an operation if performing that operation on elements within the set always results in another element within the same set. Think of it like this: if you have a club (the set) and the rule is that members can only have kids who are also club members (the operation), then the club is closed under the "having kids" operation if all the kids are indeed members. If someone has a kid who isn't a member, then the club isn't closed.

In our case, we're talking about the closure property of multiplication for polynomials. This basically means that when you multiply two polynomials together, the result should also be a polynomial. If the result is something else, like a rational function or something even weirder, then polynomials wouldn't be closed under multiplication. Understanding this fundamental concept is crucial for solving the problem at hand.

The Problem: Multiplying Polynomials and Closure

Okay, now let's tackle the problem. We're given two polynomials: $x^2 + 3x - 1$ and $2x^2 - 2x + 1$. The question asks us which statement justifies that the product of these two polynomials obeys the closure property of multiplication. We have three options to consider:

A. The result $2x^4 - 6x^2 - 1$ has a degree of 4 B. The result $2x^4 - 6x^2 - 1$ is a trinomial. C. The result $2x^4 + 4x^3 - 7x^2 + 5x - 1$

To figure this out, we first need to multiply the two polynomials together. This is where our algebra skills come into play. We'll use the distributive property, making sure each term in the first polynomial multiplies each term in the second polynomial. It's like making sure everyone at a party shakes hands with everyone else!

Multiplying the Polynomials Step-by-Step

Let's break it down:

$(x^2 + 3x - 1)(2x^2 - 2x + 1)$

First, we'll distribute $x^2$ across the second polynomial:

$x^2(2x^2 - 2x + 1) = 2x^4 - 2x^3 + x^2$

Next, we'll distribute $3x$:

$3x(2x^2 - 2x + 1) = 6x^3 - 6x^2 + 3x$

Finally, we'll distribute $-1$:

$-1(2x^2 - 2x + 1) = -2x^2 + 2x - 1$

Now, we add all these results together:

$(2x^4 - 2x^3 + x^2) + (6x^3 - 6x^2 + 3x) + (-2x^2 + 2x - 1)$

Combine like terms (those with the same exponent of $x$):

$2x^4 + (-2x^3 + 6x^3) + (x^2 - 6x^2 - 2x^2) + (3x + 2x) - 1$

This simplifies to:

$2x^4 + 4x^3 - 7x^2 + 5x - 1$

So, the result of multiplying the two polynomials is $2x^4 + 4x^3 - 7x^2 + 5x - 1$.

Analyzing the Options

Now that we have the result, let's look at the options again:

A. The result $2x^4 - 6x^2 - 1$ has a degree of 4 B. The result $2x^4 - 6x^2 - 1$ is a trinomial. C. The result $2x^4 + 4x^3 - 7x^2 + 5x - 1$

Notice that options A and B present a different result than what we calculated. This is a key observation! Option A states the result is $2x^4 - 6x^2 - 1$, and option B refers to the same result. However, our calculation shows the correct result is $2x^4 + 4x^3 - 7x^2 + 5x - 1$. Therefore, options A and B are incorrect because they're based on a wrong calculation.

Option C, on the other hand, presents the correct result of our multiplication: $2x^4 + 4x^3 - 7x^2 + 5x - 1$. This is a polynomial, which means that the product of our two original polynomials is also a polynomial. This directly demonstrates the closure property of multiplication for polynomials.

Why Option C Justifies the Closure Property

The crucial reason why option C justifies the closure property lies in the definition of a polynomial. A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Our result, $2x^4 + 4x^3 - 7x^2 + 5x - 1$, perfectly fits this definition. It's a combination of terms with $x$ raised to non-negative integer powers (4, 3, 2, 1, and 0), coefficients (2, 4, -7, 5, and -1), and operations of addition and subtraction.

Since the result of multiplying our two polynomials is itself a polynomial, it confirms that the set of polynomials is closed under multiplication. No matter which two polynomials you multiply, you'll always get another polynomial. This is the essence of the closure property.

Why Options A and B are Incorrect

Let's quickly address why options A and B don't justify the closure property, even if they were the correct result (which they aren't!).

• Option A: The result $2x^4 - 6x^2 - 1$ has a degree of 4

  While it's true that the degree of this polynomial is 4, the degree alone doesn't guarantee closure. The degree tells us the highest power of the variable, but it doesn't tell us if the result is still a polynomial. A degree of 4 simply means it's a quartic polynomial, but it doesn't inherently prove the closure property. We need to see the entire expression and confirm it fits the polynomial definition.

• Option B: The result $2x^4 - 6x^2 - 1$ is a trinomial.

  Similarly, the fact that the expression is a trinomial (meaning it has three terms) doesn't justify the closure property. Trinomials are a type of polynomial, but just knowing it's a trinomial doesn't prove that multiplying two polynomials always results in a polynomial. It's a descriptive characteristic, not a proof of closure.

Key Takeaways

So, to recap, the correct answer is C because the result, $2x^4 + 4x^3 - 7x^2 + 5x - 1$, is indeed a polynomial. This directly demonstrates the closure property of multiplication for polynomials. The other options are incorrect because they either present the wrong result or don't provide a sufficient justification for the closure property.

In summary, when we multiply two polynomials, the result is always another polynomial. This is the closure property in action!

I hope this explanation clears things up for you guys. Polynomials and the closure property can seem tricky at first, but with a little practice, they become much easier to understand. Keep up the great work!

Conclusion: The Beauty of Closure

The closure property is a fundamental concept in mathematics, and understanding it in the context of polynomials gives us a deeper appreciation for how these expressions behave. By correctly multiplying the polynomials and recognizing that the result is also a polynomial, we've successfully demonstrated the closure property of multiplication. Remember, it's not just about getting the right answer; it's about understanding why that answer is correct. Keep exploring the fascinating world of mathematics, and you'll discover even more of these beautiful connections!