1. Complete the table to summarize each student's conjecture about how to solve the problem. (2 points: 1 point for each row of the chart)
\begin{tabular}{|c|c|}
\hline Classmate & Conjecture \\
\hline Emily & Emily believes that by multiplying the two polynomials together she can compare the two resulting expressions to see if they are identical. \\
\hline Zach & Zach thinks simplifying each polynomial product on both sides and then comparing the simplified forms will show if the original products are equal. \\
\hline
\end{tabular}
Now let's proceed to the step-by-step solution:
Step-by-Step Solution:
Polynomial Product A: [tex]\((4x^2 - 4x)(x^2 - 4)\)[/tex]
1. Distribute [tex]\(4x^2\)[/tex] into [tex]\((x^2 - 4)\)[/tex]:
[tex]\[
4x^2 \cdot x^2 + 4x^2 \cdot (-4)
= 4x^4 - 16x^2
\][/tex]
2. Distribute [tex]\(-4x\)[/tex] into [tex]\((x^2 - 4)\)[/tex]:
[tex]\[
-4x \cdot x^2 + (-4x) \cdot (-4)
= -4x^3 + 16x
\][/tex]
3. Combine like terms:
[tex]\[
4x^4 - 4x^3 - 16x^2 + 16x
\][/tex]
So, after multiplying, Polynomial Product A simplifies to:
[tex]\[
4x^4 - 4x^3 - 16x^2 + 16x
\][/tex]
Polynomial Product B: [tex]\((x^2 + x - 2)(4x^2 - 8x)\)[/tex]
1. Distribute [tex]\(x^2\)[/tex] into [tex]\((4x^2 - 8x)\)[/tex]:
[tex]\[
x^2 \cdot 4x^2 + x^2 \cdot (-8x)
= 4x^4 - 8x^3
\][/tex]
2. Distribute [tex]\(x\)[/tex] into [tex]\((4x^2 - 8x)\)[/tex]:
[tex]\[
x \cdot 4x^2 + x \cdot (-8x)
= 4x^3 - 8x^2
\][/tex]
3. Distribute [tex]\(-2\)[/tex] into [tex]\((4x^2 - 8x)\)[/tex]:
[tex]\[
-2 \cdot 4x^2 + (-2) \cdot (-8x)
= -8x^2 + 16x
\][/tex]
4. Combine like terms:
[tex]\[
4x^4 - 8x^3 + 4x^3 - 8x^2 - 8x^2 + 16x
\][/tex]
5. Simplify:
[tex]\[
4x^4 - 4x^3 - 16x^2 + 16x
\][/tex]
So, after multiplying, Polynomial Product B simplifies to:
[tex]\[
4x^4 - 4x^3 - 16x^2 + 16x
\][/tex]
Conclusion and Summary for Emily and Zach:
Both polynomial products A and B simplify to the same expression:
[tex]\[
4x^4 - 4x^3 - 16x^2 + 16x
\][/tex]
Hence, Emily and Zach can clearly see that the products of the two polynomials are indeed the same.
