5. Which of the following shows that polynomials are closed under addition when two polynomials [tex]4x + 6[/tex] and [tex]2x^2 - 8x - 4[/tex] are added?

A. [tex]2x^2 - 4x + 2[/tex] will be a polynomial.
B. [tex]2x^2 - 4x + 2[/tex] may or may not be a polynomial.
C. [tex]2x^2 - 12x - 10[/tex] will be a polynomial.
D. [tex]2x^2 - 12x - 10[/tex] may or may not be a polynomial.



Answer :

To determine if polynomials are closed under addition, we need to add the two given polynomials and check the result.

The two polynomials given are [tex]\(4x + 6\)[/tex] and [tex]\(2x^2 - 8x - 4\)[/tex].

First, represent each polynomial with all coefficients, including zeros for any missing terms:
1. [tex]\(4x + 6\)[/tex] can be represented as [tex]\([0, 4, 6]\)[/tex] which corresponds to [tex]\(0x^2 + 4x + 6\)[/tex].
2. [tex]\(2x^2 - 8x - 4\)[/tex] can be represented as [tex]\([2, -8, -4]\)[/tex], which is already in standard form.

Next, add these polynomials by adding the corresponding coefficients:
- The constant term: [tex]\(6 + (-4) = 2\)[/tex].
- The linear term: [tex]\(4 + (-8) = -4\)[/tex].
- The quadratic term: [tex]\(0 + 2 = 2\)[/tex].

Thus, the resulting polynomial is:
[tex]\[2x^2 - 4x + 2.\][/tex]

Now, let's review the provided options:
1. [tex]\(2x^2 - 4x + 2\)[/tex] will be a polynomial.
2. [tex]\(2x^2 - 4x + 2\)[/tex] may or may not be a polynomial.
3. [tex]\(2x^2 - 12x - 10\)[/tex] will be a polynomial.
4. [tex]\(2x^2 - 12x - 10\)[/tex] may or may not be a polynomial.

The correct result of the addition is [tex]\(2x^2 - 4x + 2\)[/tex], which is indeed a polynomial. Hence, the correct option is:

[tex]\(2x^2 - 4x + 2\)[/tex] will be a polynomial.