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.
