Let's carefully examine the situation step-by-step to understand where Lucia made the error.
1. Define the Polynomials:
The two polynomials given are:
[tex]\[
\text{poly1} = 3x^2 + 3x + 5
\][/tex]
[tex]\[
\text{poly2} = 7x^2 - 9x + 8
\][/tex]
2. Combine Like Terms:
For each type of term (i.e., [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant), we need to combine the coefficients:
- [tex]\(x^2\)[/tex] terms:
[tex]\[
3x^2 + 7x^2 = 10x^2
\][/tex]
This is correct.
- [tex]\(x\)[/tex] terms:
[tex]\[
3x + (-9x) = -6x
\][/tex]
Lucia’s result was [tex]\(-12x\)[/tex]. Here, we observe a discrepancy.
- Constant terms:
[tex]\[
5 + 8 = 13
\][/tex]
This is correct.
3. Check Lucia’s Answer:
Lucia provided the result:
[tex]\[
10x^2 - 12x + 13
\][/tex]
Comparing her result to our combined terms:
[tex]\[
\text{Correct: } 10x^2 - 6x + 13
\][/tex]
[tex]\[
\text{Lucia: } 10x^2 - 12x + 13
\][/tex]
4. Identify the Errors:
Looking closely, we notice:
- The [tex]\(x^2\)[/tex] term is correctly combined: [tex]\(10x^2\)[/tex].
- The constant term is correctly combined: [tex]\(13\)[/tex].
- The [tex]\(x\)[/tex] term is incorrectly combined. The correct [tex]\(x\)[/tex] term should be [tex]\(-6x\)[/tex], but Lucia has [tex]\(-12x\)[/tex].
Error Analysis:
- [tex]\(x\)[/tex] term discrepancy: Lucia combined the terms [tex]\(3x\)[/tex] and [tex]\(-9x\)[/tex] incorrectly, resulting in [tex]\(-12x\)[/tex] instead of [tex]\(-6x\)[/tex].
Therefore, the correct answer to the question is:
She combined the terms [tex]\(3x\)[/tex] and [tex]\(-9x\)[/tex] incorrectly.
