Let's solve the given problem step by step to determine which of the given choices correctly shows that polynomials are closed under addition when adding the polynomials [tex]\(4x^2 - 8x - 7\)[/tex] and [tex]\(-5x + 16\)[/tex].
### Step-by-Step Solution:
1. Given Polynomials:
- Polynomial 1: [tex]\(4x^2 - 8x - 7\)[/tex]
- Polynomial 2: [tex]\(-5x + 16\)[/tex]
2. Align Polynomials by Degree:
We can rewrite the polynomials to align terms of the same degree together:
- Polynomial 1, [tex]\(4x^2 - 8x - 7\)[/tex], remains the same because it already includes all degrees up to [tex]\(x^2\)[/tex].
- Polynomial 2, [tex]\(-5x + 16\)[/tex], can be rewritten to include the missing degrees:
[tex]\[
0x^2 - 5x + 16
\][/tex]
3. Add the Corresponding Coefficients:
- Coefficients of [tex]\(x^2\)[/tex]: [tex]\(4 + 0 = 4\)[/tex]
- Coefficients of [tex]\(x\)[/tex]: [tex]\(-8 + (-5) = -13\)[/tex]
- Constant terms: [tex]\(-7 + 16 = 9\)[/tex]
4. Form the Resulting Polynomial:
Combining the results gives us:
[tex]\[
4x^2 - 13x + 9
\][/tex]
5. Verify the Resulting Polynomial:
- Degrees: The polynomial [tex]\(4x^2 - 13x + 9\)[/tex] is a valid polynomial because it contains terms with integer coefficients and maintains the polynomial structure.
- Therefore, [tex]\(4x^2 - 13x + 9\)[/tex] is indeed a polynomial.
6. Determine the Correct Answer:
- Among the given choices:
- Choice 1: [tex]\(4x^2 - 13x + 9\)[/tex], may or may not be a polynomial
- Choice 2: [tex]\(4x^2 + 13x - 23\)[/tex]; may or may not be a polynomial
- Choice 3: [tex]\(4x^2 - 13x + 9\)[/tex], will be a polynomial (This one is correct based on verification)
- Choice 4: [tex]\(4x^2 + 13x - 23\)[/tex], will be a polynomial
### Conclusion:
The correct answer is:
[tex]\[ \boxed{4x^2 - 13x + 9 \text{, will be a polynomial}} \][/tex]
This shows that polynomials are closed under addition.
