To determine if Cherise used algebra tiles to correctly represent the product of [tex]\((x-2)(x-3)\)[/tex], let's go through the steps of expanding and simplifying this expression manually.
### Step-by-Step Expansion
1. Write the expression:
[tex]\[
(x-2)(x-3)
\][/tex]
2. Apply the distributive property (also known as the FOIL method for binomials):
- First: [tex]\( x \cdot x \)[/tex]
- Outer: [tex]\( x \cdot (-3) \)[/tex]
- Inner: [tex]\((-2) \cdot x \)[/tex]
- Last: [tex]\((-2) \cdot (-3) \)[/tex]
3. Perform each multiplication:
- [tex]\( x \cdot x = x^2 \)[/tex]
- [tex]\( x \cdot (-3) = -3x \)[/tex]
- [tex]\((-2) \cdot x = -2x \)[/tex]
- [tex]\((-2) \cdot (-3) = 6 \)[/tex]
4. Combine all resulting terms:
[tex]\[
x^2 - 3x - 2x + 6
\][/tex]
5. Combine the like terms:
- Combine the [tex]\( x \)[/tex] terms: [tex]\(-3x\)[/tex] and [tex]\(-2x\)[/tex]:
[tex]\[
-3x - 2x = -5x
\][/tex]
### Final Simplified Expression
By combining all the terms together, the final expression is:
[tex]\[
x^2 - 5x + 6
\][/tex]
### Evaluating Cherise's Representation
Cherise's goal is to represent the product as [tex]\(x^2 - 5x + 6\)[/tex]. According to the correct expansion:
1. Quadratic Term: [tex]\( x^2 \)[/tex] is correct.
2. Linear Term: [tex]\(-5x\)[/tex] is correct.
3. Constant Term: [tex]\( 6 \)[/tex] is correct.
Therefore, the final expression Cherise should have reached matches the result we calculated. Cherise correctly represented the product of [tex]\((x-2)(x-3)\)[/tex] as:
[tex]\[
x^2 - 5x + 6
\][/tex]
### Conclusion
Yes, the product is correctly represented by [tex]\( x^2 - 5x + 6 \)[/tex]. Thus, Cherise's representation is accurate.
