To determine the terms that complete the factorization of the quadratic expression [tex]\(x^2 + 27x + 162\)[/tex], let's go through the process step-by-step.
1. Identify the quadratic expression: The expression given is [tex]\(x^2 + 27x + 162\)[/tex].
2. Factoring the quadratic expression: We need to find two numbers that multiply together to give the constant term, which is 162, and add up to the coefficient of the linear term, which is 27.
3. Finding Factors: The two numbers that meet these criteria are 9 and 18. This is because:
- [tex]\(9 \times 18 = 162\)[/tex]
- [tex]\(9 + 18 = 27\)[/tex]
4. Expressing the linear term: We can rewrite the middle term, 27x, using these two numbers: [tex]\(27x = 9x + 18x\)[/tex].
5. Verification: Now we can check if these terms correctly represent the quadratic expression:
- Original expression: [tex]\(x^2 + 27x + 162\)[/tex]
- Rewritten using factors: [tex]\(x^2 + 9x + 18x + 162\)[/tex]
6. Matching with given options: The terms that appear in our factored format are:
- The constant middle term: [tex]\(27\)[/tex]
- The separated linear terms: [tex]\(9x \text{ and } 18x\)[/tex]
Given these steps, the terms that complete the factorization of [tex]\(x^2 + 27x + 162\)[/tex] represented by the model are indeed:
[tex]\[27, \, 9x, \, 18x\][/tex]
Hence, the correct answer is:
[tex]\[ 27, \, 9x, \, 18x \][/tex]
