To determine which equation is represented by the polynomial model, we need to expand each of the given factor pairs (options) and see which one correctly matches the polynomial model.
1. Option 1: [tex]\((x - 3)(x + 1)\)[/tex]:
- Expand the factors:
[tex]\[
(x - 3)(x + 1) = x(x + 1) - 3(x + 1) = x^2 + x - 3x - 3 = x^2 - 2x - 3
\][/tex]
- This polynomial is [tex]\(x^2 - 2x - 3\)[/tex].
2. Option 2: [tex]\((x - 3)(x - 1)\)[/tex]:
- Expand the factors:
[tex]\[
(x - 3)(x - 1) = x(x - 1) - 3(x - 1) = x^2 - x - 3x + 3 = x^2 - 4x + 3
\][/tex]
- This polynomial is [tex]\(x^2 - 4x + 3\)[/tex].
3. Option 3: [tex]\((x + 3)(x - 1)\)[/tex]:
- Expand the factors:
[tex]\[
(x + 3)(x - 1) = x(x - 1) + 3(x - 1) = x^2 - x + 3x - 3 = x^2 + 2x - 3
\][/tex]
- This polynomial is [tex]\(x^2 + 2x - 3\)[/tex].
4. Option 4: [tex]\((x + 3)(x + 1)\)[/tex]:
- Expand the factors:
[tex]\[
(x + 3)(x + 1) = x(x + 1) + 3(x + 1) = x^2 + x + 3x + 3 = x^2 + 4x + 3
\][/tex]
- This polynomial is [tex]\(x^2 + 4x + 3\)[/tex].
By matching these expanded polynomials to the given polynomial [tex]\(x^2 - 2x - 3\)[/tex], we see that:
- Option 1: [tex]\(x^2 - 2x - 3\)[/tex] matches the given polynomial.
Therefore, the correct equation represented by the model is:
[tex]\[ x^2 - 2x - 3 = (x - 3)(x + 1) \][/tex]
Thus, the correct answer is:
[tex]\[ x^2 - 2x - 3 = (x - 3)(x + 1) \][/tex]
