Question
Use the following multiplication problem to show how multiplying polynomials is a closed system.
(x2−1)(−2x+5)
(2 points)
Answers
Answered by
GPT-4o mini
To show that multiplying polynomials is a closed system, we can evaluate the multiplication of the given polynomials \((x^2 - 1)(-2x + 5)\). The result should also be a polynomial.
Let's perform the multiplication step by step:
1. **Distributing each term of the first polynomial to each term of the second polynomial**:
\[
(x^2 - 1)(-2x + 5) = x^2(-2x) + x^2(5) - 1(-2x) - 1(5)
\]
Now, we will calculate each term:
- \(x^2(-2x) = -2x^3\)
- \(x^2(5) = 5x^2\)
- \(-1(-2x) = 2x\)
- \(-1(5) = -5\)
2. **Putting it all together**:
\[
-2x^3 + 5x^2 + 2x - 5
\]
3. **Combine the results**:
\[
-2x^3 + 5x^2 + 2x - 5
\]
As we can see, the result of the multiplication is:
\[
-2x^3 + 5x^2 + 2x - 5
\]
This expression is also a polynomial.
### Conclusion:
The multiplication of two polynomials \((x^2 - 1)\) and \((-2x + 5)\) resulted in another polynomial \(-2x^3 + 5x^2 + 2x - 5\). This demonstrates that the set of polynomials is closed under multiplication since the result of multiplying two polynomials is still a polynomial.
Let's perform the multiplication step by step:
1. **Distributing each term of the first polynomial to each term of the second polynomial**:
\[
(x^2 - 1)(-2x + 5) = x^2(-2x) + x^2(5) - 1(-2x) - 1(5)
\]
Now, we will calculate each term:
- \(x^2(-2x) = -2x^3\)
- \(x^2(5) = 5x^2\)
- \(-1(-2x) = 2x\)
- \(-1(5) = -5\)
2. **Putting it all together**:
\[
-2x^3 + 5x^2 + 2x - 5
\]
3. **Combine the results**:
\[
-2x^3 + 5x^2 + 2x - 5
\]
As we can see, the result of the multiplication is:
\[
-2x^3 + 5x^2 + 2x - 5
\]
This expression is also a polynomial.
### Conclusion:
The multiplication of two polynomials \((x^2 - 1)\) and \((-2x + 5)\) resulted in another polynomial \(-2x^3 + 5x^2 + 2x - 5\). This demonstrates that the set of polynomials is closed under multiplication since the result of multiplying two polynomials is still a polynomial.