Explain why the following operation is a close system under subtraction
(9x^2 + 17x +13)- 2x^2+12x+11
The given operation is subtraction, and we need to determine whether it forms a closed system.
To show that subtraction is a closed operation, we need to demonstrate that subtracting any two polynomials will always result in a polynomial.
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 subtract corresponding coefficients of like terms.
Subtracting the constants: 11 - 13 = -2
Subtracting the coefficients of x: 12x - 17x = -5x
Subtracting the coefficients of x^2: 2x^2 - 9x^2 = -7x^2
Therefore, the result of the subtraction is the polynomial -7x^2 - 5x - 2, which is a polynomial and confirms that subtraction is a closed operation in this case.
To determine if an operation is a closed system under subtraction, we need to perform the operation and check if the result belongs to the same set as the operands. In this case, we are subtracting (9x^2 + 17x + 13) from (2x^2 + 12x + 11).
Let's perform the subtraction:
(9x^2 + 17x + 13) - (2x^2 + 12x + 11)
First, distribute the negative sign to each term in the parentheses:
9x^2 + 17x + 13 -2x^2 - 12x - 11
Now, simplify the expression by combining like terms:
(9x^2 - 2x^2) + (17x - 12x) + (13 - 11)
7x^2 + 5x + 2
The resulting expression is 7x^2 + 5x + 2.
Since the result is still a quadratic expression (an equation with the highest power of x being 2) in terms of x, it belongs to the same set as the original expression. Therefore, the operation (subtraction) is a closed system in this case.
To understand why the operation (9x^2 + 17x + 13) - (2x^2 + 12x + 11) is a closed system under subtraction, we need to define what a closed system is in mathematics.
In mathematics, a closed system is a set of elements and an operation, where performing the operation on any two elements in the set always produces a result that is also an element of the set.
In this case, the set of elements consists of all polynomials with degree less than or equal to 2 (since the largest exponent is 2 in both terms of the subtraction operation). The operation is subtraction, denoted by (-).
Looking at the given operation: (9x^2 + 17x + 13) - (2x^2 + 12x + 11), we can see that it involves subtracting two polynomials with the same degree. When subtracting each term, we combine like terms and obtain:
(9x^2 - 2x^2) + (17x - 12x) + (13 - 11)
Simplifying this expression, we have:
7x^2 + 5x + 2
Now, we need to check if this resulting polynomial belongs to the set of polynomials with degree less than or equal to 2. Since the highest degree of the polynomial is 2, it meets this requirement and is an element of the set.
Therefore, the operation (9x^2 + 17x + 13) - (2x^2 + 12x + 11) is a closed system under subtraction because it produces a polynomial that belongs to the set of polynomials with degree less than or equal to 2.