In 1–2 sentences, explain why the following operation is a closed system under subtraction.
(9x^2+17x+13)−(2x^2+12x+11)
The operation of subtracting (9x^2+17x+13) from (2x^2+12x+11) is a closed system because the resulting expression will still be a polynomial, as all terms have the same degree.
How
Sorry for the confusion. The operation of subtracting (9x^2+17x+13) from (2x^2+12x+11) is a closed system under subtraction because when subtracting polynomials, the result will always be a polynomial.
Explain it
Sure, let me explain it in more detail.
In this case, we are subtracting the polynomial (9x^2+17x+13) from the polynomial (2x^2+12x+11). When we subtract these two polynomials, we combine like terms. Each term is a product of a coefficient and a variable raised to a power.
For example, when subtracting the x^2 terms, we have 2x^2 - 9x^2. This simplifies to -7x^2.
Similarly, when subtracting the x terms, we have 12x - 17x. This simplifies to -5x.
Lastly, when subtracting the constant terms, we have 11 - 13. This simplifies to -2.
So, the result of the subtraction is the polynomial -7x^2 - 5x - 2.
Since the result is still a polynomial with terms of the same degree, it is a closed system under subtraction.
Use the following multiplication problem to show how multiplying polynomials is a closed system. (x^2−1)(−2x+5)
To show that multiplying polynomials is a closed system, we can expand the given expression (x^2−1)(−2x+5) using the distributive property.
(x^2 - 1)(-2x + 5)
= x^2*(-2x) + x^2*5 + (-1)*(-2x) + (-1)*5
= -2x^3 + 5x^2 + 2x - 5
As we can see, the result of multiplying these two polynomials is also a polynomial (-2x^3 + 5x^2 + 2x - 5). Since the result is a polynomial, and thus part of the same system, we can conclude that multiplying polynomials is closed under this operation.