To determine if Cherise has correctly represented the product of [tex]\((x-2)(x-3)\)[/tex], we need to calculate the product step by step and compare it with her work. Let's go through the multiplication process:
### 1. Distribute [tex]\(x\)[/tex] across each term in [tex]\((x-3)\)[/tex]:
[tex]\[
x(x - 3) = x^2 - 3x
\][/tex]
### 2. Distribute [tex]\(-2\)[/tex] across each term in [tex]\((x-3)\)[/tex]:
[tex]\[
-2(x - 3) = -2x + 6
\][/tex]
### 3. Combine all the terms together:
[tex]\[
(x^2 - 3x) + (-2x + 6) = x^2 - 3x - 2x + 6
\][/tex]
### 4. Simplify by combining like terms:
[tex]\[
x^2 - 5x + 6
\][/tex]
Now let's compare this to Cherise's representation. The table Cherise provided indicates:
- For [tex]\(+x \cdot +x\)[/tex], she has [tex]\(+x^2\)[/tex], which is correct.
- For [tex]\(+x \cdot -2\)[/tex], she has [tex]\(-2x\)[/tex], which is correct.
- For [tex]\(-2 \cdot +x\)[/tex], she also has [tex]\(-2x\)[/tex], which is correct.
- For [tex]\(-2 \cdot -3\)[/tex], this should be a positive term ([tex]\(+6\)[/tex]), since multiplying two negative numbers results in a positive product.
Given these calculations:
1. Cherise accounted for the multiplication of the [tex]\(x\)[/tex]-tiles by the negative integer tiles correctly.
2. She correctly multiplied the negative integer tiles.
3. However, she made an error in the sign and value combination when adding the terms, as it resulted in [tex]\(-x - x - x\)[/tex], simplifying incorrectly to [tex]\(-5x\)[/tex] without accounting for all terms correctly.
4. The correct expanded form of [tex]\((x-2)(x-3)\)[/tex] should be [tex]\(x^2 - 5x + 6\)[/tex].
### Correct Response:
No, Cherise did not add the terms together correctly. The correct product should be [tex]\(x^2 - 5x + 6\)[/tex].
