Duplicate Question
The question on this page has been marked as a duplicate question.
Original Question
Use the following multiplication problem to show how multiplying polynomials is a closed system. (x2−1)(−2x+5)Question
Use the following multiplication problem to show how multiplying polynomials is a closed system...
(x^2−1)(−2x+5)
(x^2−1)(−2x+5)
Answers
Answer
shorten
Answered by
GPT-4o mini
To show that multiplying polynomials is a closed system, we will perform the multiplication of the polynomials \((x^2 - 1)\) and \((-2x + 5)\) and demonstrate that the result is also a polynomial.
Let's multiply the two polynomials step by step:
### Step 1: Apply the Distributive Property (FOIL)
We will distribute each term in the first polynomial to each term in the second polynomial:
\[
(x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 + (-1) \cdot (-2x) + (-1) \cdot 5
\]
### Step 2: Calculate Each Term
Now, let’s compute each of these products:
1. \(x^2 \cdot (-2x) = -2x^3\)
2. \(x^2 \cdot 5 = 5x^2\)
3. \((-1) \cdot (-2x) = 2x\)
4. \((-1) \cdot 5 = -5\)
### Step 3: Combine All the Terms
Now combine all these results:
\[
-2x^3 + 5x^2 + 2x - 5
\]
### Result
The multiplication of \((x^2 - 1)\) and \((-2x + 5)\) results in:
\[
-2x^3 + 5x^2 + 2x - 5
\]
### Conclusion
The result \(-2x^3 + 5x^2 + 2x - 5\) is clearly a polynomial.
This demonstrates that the system is closed under multiplication, as the product of two polynomials is itself a polynomial. Thus, multiplying polynomials does not produce an outcome outside the set of polynomials, confirming the closure property of polynomial multiplication.
Let's multiply the two polynomials step by step:
### Step 1: Apply the Distributive Property (FOIL)
We will distribute each term in the first polynomial to each term in the second polynomial:
\[
(x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 + (-1) \cdot (-2x) + (-1) \cdot 5
\]
### Step 2: Calculate Each Term
Now, let’s compute each of these products:
1. \(x^2 \cdot (-2x) = -2x^3\)
2. \(x^2 \cdot 5 = 5x^2\)
3. \((-1) \cdot (-2x) = 2x\)
4. \((-1) \cdot 5 = -5\)
### Step 3: Combine All the Terms
Now combine all these results:
\[
-2x^3 + 5x^2 + 2x - 5
\]
### Result
The multiplication of \((x^2 - 1)\) and \((-2x + 5)\) results in:
\[
-2x^3 + 5x^2 + 2x - 5
\]
### Conclusion
The result \(-2x^3 + 5x^2 + 2x - 5\) is clearly a polynomial.
This demonstrates that the system is closed under multiplication, as the product of two polynomials is itself a polynomial. Thus, multiplying polynomials does not produce an outcome outside the set of polynomials, confirming the closure property of polynomial multiplication.
Answered by
GPT-4o mini
To show that multiplying polynomials is a closed system, let's multiply the polynomials \( (x^2 - 1) \) and \( (-2x + 5) \).
### Step 1: Distribute Each Term
Using the distributive property:
\[
(x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 + (-1) \cdot (-2x) + (-1) \cdot 5
\]
### Step 2: Calculate Each Product
1. \( x^2 \cdot (-2x) = -2x^3 \)
2. \( x^2 \cdot 5 = 5x^2 \)
3. \( (-1) \cdot (-2x) = 2x \)
4. \( (-1) \cdot 5 = -5 \)
### Step 3: Combine Results
Combining all terms:
\[
-2x^3 + 5x^2 + 2x - 5
\]
### Conclusion
The result, \(-2x^3 + 5x^2 + 2x - 5\), is a polynomial, demonstrating that the multiplication of polynomials is closed under this operation.
### Step 1: Distribute Each Term
Using the distributive property:
\[
(x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 + (-1) \cdot (-2x) + (-1) \cdot 5
\]
### Step 2: Calculate Each Product
1. \( x^2 \cdot (-2x) = -2x^3 \)
2. \( x^2 \cdot 5 = 5x^2 \)
3. \( (-1) \cdot (-2x) = 2x \)
4. \( (-1) \cdot 5 = -5 \)
### Step 3: Combine Results
Combining all terms:
\[
-2x^3 + 5x^2 + 2x - 5
\]
### Conclusion
The result, \(-2x^3 + 5x^2 + 2x - 5\), is a polynomial, demonstrating that the multiplication of polynomials is closed under this operation.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.