Isabella292008idn
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Which of the following shows that polynomials are closed under subtraction when two polynomials, [tex]\left(5x^2 + 2x - 6\right) - \left(3x^2 - 6x + 2\right)[/tex], are subtracted?

A. [tex]2x^2 - 4x - 4[/tex] will be a polynomial
B. [tex]2x^2 - 4x - 4[/tex] may or may not be a polynomial
C. [tex]2x^2 + 8x - 8[/tex] will be a polynomial
D. [tex]2x^2 + 8x - 8[/tex] may or may not be a polynomial

Sagot :

GinnyAnsweridn
Let's go through the subtraction of the two given polynomials step-by-step. The polynomials we're dealing with are:

[tex]\[ P_1(x) = 5x^2 + 2x - 6 \][/tex]
[tex]\[ P_2(x) = 3x^2 - 6x + 2 \][/tex]

We need to subtract [tex]\( P_2(x) \)[/tex] from [tex]\( P_1(x) \)[/tex]:

[tex]\[ P(x) = P_1(x) - P_2(x) = (5x^2 + 2x - 6) - (3x^2 - 6x + 2) \][/tex]

To perform the subtraction, we subtract the corresponding coefficients of each term:

- For the [tex]\( x^2 \)[/tex] term:
[tex]\[ 5x^2 - 3x^2 = 2x^2 \][/tex]

- For the [tex]\( x \)[/tex] term:
[tex]\[ 2x - (-6x) = 2x + 6x = 8x \][/tex]

- For the constant term:
[tex]\[ -6 - 2 = -8 \][/tex]

Thus, after performing the subtraction, the resulting polynomial is:

[tex]\[ P(x) = 2x^2 + 8x - 8 \][/tex]

Since all the steps have led us to another polynomial, the result [tex]\( P(x) = 2x^2 + 8x - 8 \)[/tex] is indeed a polynomial and confirms that the subtraction of these two given polynomials results in another polynomial.

Therefore, among the given choices, the correct answer is:

[tex]\[ 2x^2 + 8x - 8 \text{, will be a polynomial} \][/tex]
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