The operation is a closed system under subtraction because subtracting two polynomials results in another polynomial, which remains within the same set of polynomial functions. In this case, subtracting \( (2x^2 + 12x + 11) \) from \( (9x^2 + 17x + 13) \) yields a polynomial of degree at most 2, thus ensuring closure.
In 1–2 sentences, explain why the following operation is a closed system under subtraction.
(9x2+17x+13)−(2x2+12x+11
3 answers
Use the following multiplication problem to show how multiplying polynomials is a closed system.
(x2−1)(−2x+5)
(x2−1)(−2x+5)
To demonstrate that multiplying polynomials is a closed system, let's multiply the two given polynomials: \( (x^2 - 1)(-2x + 5) \).
Using the distributive property (also known as the FOIL method for binomials), we multiply each term in the first polynomial by each term in the second polynomial:
- \( x^2 \cdot -2x = -2x^3 \)
- \( x^2 \cdot 5 = 5x^2 \)
- \( -1 \cdot -2x = 2x \)
- \( -1 \cdot 5 = -5 \)
Now, we combine all these results:
\[ -2x^3 + 5x^2 + 2x - 5 \]
The result, \( -2x^3 + 5x^2 + 2x - 5 \), is a polynomial of degree 3, which fits within the set of polynomial functions. Thus, we see that the product of the two polynomials is another polynomial, confirming that multiplication of polynomials is a closed system.