To determine the error in Chin's chart, let's break down the multiplication of the two polynomials step by step:
Given the expression:
[tex]\[
(3x + y)(x^2 + 2y + 4)
\][/tex]
We will distribute each term in the first polynomial [tex]\((3x + y)\)[/tex] to every term in the second polynomial [tex]\((x^2 + 2y + 4)\)[/tex].
### Term-by-Term Multiplication:
1. First Term ([tex]\(3x\)[/tex] multiplying each term in [tex]\((x^2 + 2y + 4)\)[/tex]):
- [tex]\(3x \cdot x^2 = 3x^3\)[/tex]
- [tex]\(3x \cdot 2y = 6xy\)[/tex]
- [tex]\(3x \cdot 4 = 12x\)[/tex]
2. Second Term ([tex]\(y\)[/tex] multiplying each term in [tex]\((x^2 + 2y + 4)\)[/tex]):
- [tex]\(y \cdot x^2 = x^2y\)[/tex]
- [tex]\(y \cdot 2y = 2y^2\)[/tex]
- [tex]\(y \cdot 4 = 4y\)[/tex]
### Now, comparing the correct row-by-row results with the entries in the chart:
| & [tex]\(x^2\)[/tex] & [tex]\(2y\)[/tex] & 4 \\
|---|---|---|---|
| [tex]\(3x\)[/tex] | [tex]\(3x^3\)[/tex] | [tex]\(6xy\)[/tex] | [tex]\(12x\)[/tex] |
| [tex]\(y\)[/tex] | [tex]\(x^2y\)[/tex] | [tex]\(2y^2\)[/tex] | [tex]\(4y\)[/tex] |
### Observations based on the chart provided:
1. The entry in the second row and second column of the chart is [tex]\(6x\)[/tex] where it should be [tex]\(6xy\)[/tex].
2. The terms in the multiplication table should be:
- [tex]\(3x^3\)[/tex] in the top row, left-most entry (which is correct).
- [tex]\(6xy\)[/tex] in the top row, middle entry.
- [tex]\(12x\)[/tex] in the top row, right-most entry (which is correct).
- [tex]\(x^2y\)[/tex] in the bottom row, left-most entry (which is correct).
- [tex]\(2y^2\)[/tex] in the bottom row, middle entry (which is correct).
- [tex]\(4y\)[/tex] in the bottom row, right-most entry (which is correct).
### Conclusion:
The correction that needs to be made to the chart is changing the term:
"The expression [tex]\(6x\)[/tex] should be [tex]\(6xy\)[/tex]."
So, the correct change to be made to the chart is:
[tex]\[
\boxed{The\ expression\ 6x\ should\ be\ 6xy.}
\][/tex]
