Question
Use the following multiplication problem to show how multiplying polynomials is a closed system.
(X^2 - 1) (-2x + 5)
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(X^2 - 1) (-2x + 5)
Make it short
Answers
Answered by
GPT-4o mini
To show that multiplying polynomials is a closed system, we will multiply \((X^2 - 1)\) and \((-2x + 5)\):
\[
(X^2 - 1)(-2x + 5) = X^2(-2x) + X^2(5) - 1(-2x) - 1(5)
\]
Calculating each term:
1. \(X^2(-2x) = -2x^3\)
2. \(X^2(5) = 5x^2\)
3. \(-1(-2x) = 2x\)
4. \(-1(5) = -5\)
Now, combine all these results:
\[
-2x^3 + 5x^2 + 2x - 5
\]
The result \(-2x^3 + 5x^2 + 2x - 5\) is still a polynomial. Since we started with polynomials and ended with a polynomial, this demonstrates that the set of polynomials is closed under multiplication.
\[
(X^2 - 1)(-2x + 5) = X^2(-2x) + X^2(5) - 1(-2x) - 1(5)
\]
Calculating each term:
1. \(X^2(-2x) = -2x^3\)
2. \(X^2(5) = 5x^2\)
3. \(-1(-2x) = 2x\)
4. \(-1(5) = -5\)
Now, combine all these results:
\[
-2x^3 + 5x^2 + 2x - 5
\]
The result \(-2x^3 + 5x^2 + 2x - 5\) is still a polynomial. Since we started with polynomials and ended with a polynomial, this demonstrates that the set of polynomials is closed under multiplication.