Fill in the missing values to show that polynomials form a closed system under subtraction.
9x2 + $$x + 13
– 2x2 + 12x + $$
$$x2 + 5x + 2
9x2 - 2x2 = 7x2
$x - 12x = -11x
$13 - 2 = 11
So the filled-in values are:
9x2 + x + 13
– 2x2 + 12x + 2
= 7x2 - 11x + 11
wrong
I apologize for the mistake. Let's correct it:
9x^2 - 2x^2 = 7x^2
x - 12x = -11x
13 - 11 = 2
So, the correct values are:
9x^2 + x + 13
- 2x^2 + 12x + 11
= 7x^2 - 11x + 2
9x^2 + (?)x + 13
To show that polynomials form a closed system under subtraction, we need to subtract one polynomial from another and verify that the result is still a polynomial.
Let's subtract the second polynomial from the first polynomial:
(9x^2 + $$x + 13) - (-2x^2 + 12x + $$)
= 9x^2 + $$x + 13 + 2x^2 - 12x - $$
= (9x^2 + 2x^2) + ($$x - 12x) + (13 - $$)
Simplifying each term:
= 11x^2 - 11x + (13 - $$)
Since each term is a polynomial with real coefficients, we can see that the result is still a polynomial. Therefore, we have shown that polynomials form a closed system under subtraction.
To show that polynomials form a closed system under subtraction, we need to subtract each term of the second polynomial from the corresponding term of the first polynomial.
Given polynomials:
1) 9x^2 + $$x + 13
2) -2x^2 + 12x + $$
Let's subtract the coefficients of each term:
1) For the x^2 term, 9x^2 - (-2x^2) = 9x^2 + 2x^2 = 11x^2
2) For the x term, $$x - 12x = (whatever the value of x is) - 12x = ($$ - 12)x = -12x
3) For the constant term, 13 - $$ = (whatever the value of $$ is) - (whatever the value of $$ is) = 0
So, the resulting polynomial after subtracting the second polynomial from the first polynomial is:
11x^2 - 12x + 0
Note: The values represented by "$$" in the question were not specified, so we cannot determine their individual values. However, regardless of the specific values, we can still perform the subtraction operation to show that polynomials form a closed system under subtraction.
- 2x^2 + 12x + 2
= 7x^2 + (?)x + 11
To fill in the missing terms:
9x^2 - 2x^2 = 7x^2
For the missing term in the x coefficient: x - 12x = -11x
And for the missing term in the constant: 13 - 2 = 11
Therefore, the filled in values are:
9x^2 + (-11)x + 13
- 2x^2 + 12x + 2
= 7x^2 + (-11)x + 11