Now that you better understand a closed system, you can investigate closure, or polynomial, multiplication. Remember that this is another way of asking, “Does multiplying polynomials always result in another polynomial?” Consider the problem (−4x2)(5x3) . First, multiply the coefficients together, then the exponents. −4⋅5 ⋅x2 x3 Remember that when multiplying exponents, you add the powers. −20x5 The result is a polynomial, thus, this example is a closed system. Example 1 Consider the following polynomials that are being multiplied. For this polynomial multiplication problem, you will show how performing the operation (multiplication) to the set (of polynomials) produces members of the same set (polynomials). 2x2(y+4z−3) Solution To multiply, distribute 2x2 to each term in the parentheses. 2x2(y)+2x2(4z)−2x2(3) Next, multiply the coefficients when applicable. 2x2y+8x2z−6x2 The result is a polynomial because all the powers of the exponents are positive. Thus, this example is a closed system. Note that 2x2 , although only one term, is still a polynomial. Specifically, it is a monomial (one term). Multiplying a monomial with a polynomial is still considered a polynomial operation. It is also important to note that an expression must contain a variable and at least two terms to be considered a polynomial. Check-In Is multiplying a constant (one term) with a trinomial considered a polynomial operation, and therefore a closed set? Reveal Answer Consider another example of applying multiplication to a polynomial. Example 2 (5x3+1)(9x−2) Solution You can multiply the polynomials using the tabular method. Set up the Distributive Property using the tabular method. 9x −2 5x3 5x3⋅9x 5x3⋅−2 +1 +1⋅9x +1⋅−2 Multiply the values in each cell. 9x −2 5x3 5x3⋅9x=45x4 5x3⋅−2=10x3 +1 +1⋅9x=9x +1⋅−2=−2 Collect each term and write them horizontally to create the expression. 45x4−10x3+9x−2 The result is a polynomial, thus, this example is a closed system. Here are some more examples: (2x4)(3x2)=6x6 (x2)(x2−1)= x4−x2 (x+y)(x −y)=x2−y2 (2x−1)(3x+2)=6x2+x−2 In all of these cases, the result is a polynomial.

Question

Now that you better understand a closed system, you can investigate closure, or polynomial, multiplication. Remember that this is another way of asking, “Does multiplying polynomials always result in another polynomial?” Consider the problem (−4x2)(5x3) . First, multiply the coefficients together, then the exponents. −4⋅5 ⋅x2 x3 Remember that when multiplying exponents, you add the powers. −20x5 The result is a polynomial, thus, this example is a closed system.  Example 1 Consider the following polynomials that are being multiplied. For this polynomial multiplication problem, you will show how performing the operation (multiplication) to the set (of polynomials) produces members of the same set (polynomials). 2x2(y+4z−3) Solution To multiply, distribute 2x2 to each term in the parentheses. 2x2(y)+2x2(4z)−2x2(3) Next, multiply the coefficients when applicable. 2x2y+8x2z−6x2 The result is a polynomial because all the powers of the exponents are positive. Thus, this example is a closed system. Note that 2x2 , although only one term, is still a polynomial. Specifically, it is a monomial (one term). Multiplying a monomial with a polynomial is still considered a polynomial operation. It is also important to note that an expression must contain a variable and at least two terms to be considered a polynomial. Check-In Is multiplying a constant (one term) with a trinomial considered a polynomial operation, and therefore a closed set? Reveal Answer Consider another example of applying multiplication to a polynomial. Example 2 (5x3+1)(9x−2) Solution You can multiply the polynomials using the tabular method. Set up the Distributive Property using the tabular method. 9x −2 5x3 5x3⋅9x 5x3⋅−2 +1 +1⋅9x +1⋅−2 Multiply the values in each cell. 9x −2 5x3 5x3⋅9x=45x4 5x3⋅−2=10x3 +1 +1⋅9x=9x +1⋅−2=−2 Collect each term and write them horizontally to create the expression. 45x4−10x3+9x−2 The result is a polynomial, thus, this example is a closed system. Here are some more examples: (2x4)(3x2)=6x6 (x2)(x2−1)= x4−x2 (x+y)(x −y)=x2−y2 (2x−1)(3x+2)=6x2+x−2 In all of these cases, the result is a polynomial.

Using this info, provide me an answer to the question:
Use the following multiplication problem to show how multiplying polynomials is a closed system. (x2−1)(−2x+5) (2 points)

GPT-4o mini · Answer

To show how multiplying the polynomials \((x^2 - 1)(-2x + 5)\) demonstrates that polynomial multiplication is a closed system, we can follow the steps of the multiplication process.

Step 1: Use the Distributive Property

Distribute each term in the first polynomial \((x^2 - 1)\) to each term in the second polynomial \((-2x + 5)\):

\[ (x^2)(-2x) + (x^2)(5) + (-1)(-2x) + (-1)(5) \]

Step 2: Multiply the Terms

Now, perform the multiplication for each term:

\(x^2 \cdot (-2x) = -2x^3\)
\(x^2 \cdot 5 = 5x^2\)
\(-1 \cdot (-2x) = 2x\)
\(-1 \cdot 5 = -5\)

Putting it all together, we get:

\[ -2x^3 + 5x^2 + 2x - 5 \]

Step 3: Combine Like Terms

In this case, there are no like terms to combine, so we keep all the terms as they are:

\[ -2x^3 + 5x^2 + 2x - 5 \]

Conclusion

The result of multiplying \((x^2 - 1)(-2x + 5)\) is \(-2x^3 + 5x^2 + 2x - 5\).

Verification of Being a Polynomial

This expression is a polynomial because:

It consists of multiple terms.
Each term is a product of a constant and a variable raised to a non-negative integer exponent.
The powers of the exponents are all non-negative (0, 1, 2, 3) and there are no negative exponents.

Thus, since we started with polynomials and the operation of multiplication produced another polynomial, we conclude that multiplying polynomials is a closed system.

Answer

what is the /cdot for? Just write it like it is you know what I mean?

GPT-4o mini · Answer

Absolutely! Let's rewrite the multiplication without using the multiplication dot (•).

To show how multiplying the polynomials \((x^2 - 1)(-2x + 5)\) demonstrates that polynomial multiplication is a closed system, we'll follow these steps:

Step 1: Use the Distributive Property

Distribute each term in the first polynomial \((x^2 - 1)\) to each term in the second polynomial \((-2x + 5)\):

\[ x^2 \cdot (-2x) + x^2 \cdot 5 + (-1) \cdot (-2x) + (-1) \cdot 5 \]

Step 2: Multiply the Terms

Now, perform the multiplication for each term:

\(x^2 \cdot (-2x) = -2x^3\)
\(x^2 \cdot 5 = 5x^2\)
\(-1 \cdot (-2x) = 2x\)
\(-1 \cdot 5 = -5\)

Putting it all together, we get:

\[ -2x^3 + 5x^2 + 2x - 5 \]

Conclusion

The result of multiplying \((x^2 - 1)(-2x + 5)\) is:

\[ -2x^3 + 5x^2 + 2x - 5 \]

Verification of Being a Polynomial

This expression is a polynomial because:

It consists of multiple terms.
Each term is a product of a constant and a variable raised to a non-negative integer exponent.
The powers of the exponents are all non-negative (0, 1, 2, 3).

Thus, since we started with polynomials and the multiplication produced another polynomial, we conclude that multiplying polynomials is indeed a closed system.