To determine the missing values in the table from Shelia's polynomial division, let's examine how the division process unfolds.
1. Looking at the quotient's coefficients:
- The quotient given in the table is [tex]\( x^3 - 3x^2 + x - 5 \)[/tex].
2. Focusing on the division table:
- The first row under "Divisor" shows the division process with the term [tex]\( x \)[/tex] and its multiplication into all terms of the quotient.
- Similarly, the second row shows the result of multiplying the divisor by [tex]\(-1\)[/tex] through all the quotient terms.
Given the table format and the results found earlier ([tex]\((-5, 3, 5)\)[/tex]):
3. Identifying value calculations:
- The missing values in the table correspond to coefficients gathered from the polynomial division carried out.
### Step-by-step determination of coefficients:
[tex]\(\mathbf{A}\)[/tex] Calculation:
- From the first row, [tex]\(x-1 = \divisor\)[/tex], we multiply [tex]\(-5\)[/tex] (constant term in the quotient) and [tex]\(x\)[/tex] from the divisor:
- [tex]\(-5 \cdot x = -5x\)[/tex].
So, [tex]\(A = -5\)[/tex].
[tex]\(\mathbf{B}\)[/tex] Calculation:
- In the second row, the term from the table under [tex]\(B\)[/tex], multiplying the second term (-[tex]\((\divisor\)[/tex] term)) from the quotient coefficient ([tex]\(-3x^2\)[/tex]) by [tex]\(-1\)[/tex]):
- [tex]\((-1 \cdot -3x^2) = 3x^2\)[/tex].
So, [tex]\(B = 3x\)[/tex].
[tex]\(\mathbf{C}\)[/tex] Calculation:
- From the constant multiplication with [tex]\(-1\)[/tex] and quotient's last term ([tex]\(-5\)[/tex]):
- [tex]\((-1) \cdot (-5) = 5\)[/tex].
So, [tex]\(C = 5\)[/tex].
### Values:
- [tex]\(A = -5\)[/tex]
- [tex]\(B = 3\)[/tex]
- [tex]\(C = 5\)[/tex]
Thus, we can see which answer given matches these values. The correct choice is:
[tex]\(\boxed{-5, 3, 5}\)[/tex].
