Use the following multiplication problem to show how multiplying polynomials is a closed system.
(x2−1)(−2x+5)
To show that multiplying polynomials is a closed system, we need to demonstrate that the product of two polynomials is also a polynomial.
Given the multiplication problem (x^2 − 1)(−2x + 5), let's expand the expression using the distributive property:
= x^2*(-2x) + x^2*5 + (-1)*(-2x) + (-1)*5
= -2x^3 + 5x^2 + 2x - 5
The result of multiplying the two polynomials is -2x^3 + 5x^2 + 2x - 5, which is a polynomial. Therefore, this demonstrates that multiplying polynomials is a closed system.
To demonstrate that multiplying polynomials is a closed system, we need to show that the result of multiplying two polynomials is always another polynomial.
The given multiplication problem is (x^2−1)(−2x+5).
To calculate this, we use the distributive property of multiplication. We multiply each term from the first polynomial (x^2−1) by each term from the second polynomial (−2x+5) and then combine like terms.
Step 1: Multiply the first term of the first polynomial by each term of the second polynomial:
(x^2)(−2x) + (x^2)(5)
Step 2: Multiply the second term of the first polynomial by each term of the second polynomial:
(−1)(−2x) + (−1)(5)
Now, we simplify each term:
Step 3: Simplify the first term:
−2x^3 + 5x^2
Step 4: Simplify the second term:
2x + (−5)
Step 5: Combine the simplified terms:
−2x^3 + 5x^2 + 2x − 5
The final result after multiplying the two polynomials is the polynomial −2x^3 + 5x^2 + 2x − 5. Since the result of multiplying these polynomials is another polynomial, we have demonstrated that multiplying polynomials is a closed system.