Fill in the missing values to show that polynomials form a closed system under subtraction.
9x2 + $$x + 13
– 2x2 + 12x + $$
$$x2 + 5x + 2
To show that polynomials form a closed system under subtraction, we need to subtract the two given polynomials and determine the missing values.
First, let's subtract the x^2 terms: 9x^2 - 2x^2 = 7x^2
Next, let's subtract the x terms: x - 12x = -11x
Finally, let's subtract the constant terms: 13 - 2 = 11
So, the missing values are:
-11x (for the missing x term)
7x^2 (for the missing x^2 term)
11 (for the missing constant term)
Putting it all together, the subtracted polynomial is:
7x^2 - 11x + 11
To show that polynomials form a closed system under subtraction, we need to subtract two polynomials and show that the result is also a polynomial.
Given the polynomials:
1) 9x^2 + x + 13
2) -2x^2 + 12x + y
3) z*x^2 + 5x + 2
To subtract the second polynomial from the first, we need to subtract the corresponding terms of the polynomials. We subtract the coefficients of corresponding terms and keep the same variables and exponents.
To find the missing values y and z, we look at the coefficients of corresponding terms on the left and right sides of the equation:
For the x term:
Coefficient on the left side = 1
Coefficient on the right side = 12x
So, we have: 1 - 12 = -11
For the constant term:
Coefficient on the left side = 13
Coefficient on the right side = y
So, we have: 13 - y = -y + 13
For the x^2 term:
Coefficient on the left side = 9x^2
Coefficient on the right side = -2x^2
So, we have: 9x^2 - (-2x^2) = 9x^2 + 2x^2 = 11x^2
Therefore, the missing values are:
y = -11
z = 11
So, the final expression after subtracting the second polynomial from the first is:
9x^2 + x + 13 - (-2x^2 + 12x + y) = 11x^2 - 11x + 24
Hence, we have shown that subtracting two polynomials results in another polynomial, demonstrating that polynomials form a closed system under subtraction.
To show that polynomials form a closed system under subtraction, we need to subtract two polynomials and see if the result is still a polynomial.
Let's subtract the second polynomial from the first:
(9x^2 + x + 13) - (2x^2 + 12x + )
To subtract, we combine like terms:
9x^2 - 2x^2 = 7x^2
x - 12x = -11x
13 - = 13
Therefore, the subtraction of the two polynomials is:
7x^2 - 11x + 13
Now, let's subtract the result from the third polynomial:
(x^2 + 5x + 2) - (7x^2 - 11x + 13)
Again, we combine like terms:
x^2 - 7x^2 = -6x^2
5x - (-11x) = 5x + 11x = 16x
2 - 13 = -11
So, the final result after subtracting the third polynomial is:
-6x^2 + 16x - 11
Since the result is still a polynomial, we have shown that polynomials form a closed system under subtraction.