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**Polynomials Unit 6 - Quest**
**Algebra 1**
**Name:** Delaltia Duncan

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**Instructions:**
Show all your work clearly. Also, give the type and degree of your solution.

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**Problems 1 and 2:**
Find the area and perimeter.

1. [Illustration showing a figure with a length of 70 ft]

2. **Expression:**
\[
6x^2 + 2
\]
\[
2x^3 - 3
\]

**Area Formula:** \[A = L \cdot W\] (Foil)
\[
A = 6x^2 + 2(2x^2 - 3)
\]
**Simplify:**
\[
12x^4 - 18x^2 + 4x^2 - 6
\]
\[
12x^4 - 14x^2 - 6
\]

**Type:** Trinomial
**Degree:** 4

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**Problem 3:**
Simplify:
\[
\left(2x^2y^7\right)^3\left(3x^4y^2\right)^{-3}\left(27x^5y^{10}\right)^0
\]
\[
2^3x^6y^{21} \cdot 3^{-3}x^{-12}y^{-6}
\]
\[
= 8x^6y^{21}
\]
\[
27x^{12}y^6
\]

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**Problem 4:**
Find the area using the expression:
\[A = L \cdot W\]
**Expression:**
\[
\left(3xy^2\right)^2 \cdot \left(2x^2\right)^4
\]
**Simplify:**
\[
3^2x^2y^4 \cdot 2^4x^8
\]
\[
= 9x^2y^4 \cdot 16x^8
\]

Answer :

To find the area given the terms [tex]\( L = (3xy^2)^2 \)[/tex] and [tex]\( W = (2x^2)^4 \)[/tex], let's break down the steps:

1. Calculate [tex]\( L = (3xy^2)^2 \)[/tex]:

- Start by squaring each component within the parentheses.
- [tex]\( 3^2 = 9 \)[/tex].
- [tex]\( x^2 = x^2 \)[/tex].
- [tex]\( (y^2)^2 = y^{2 \times 2} = y^4 \)[/tex].

Thus, [tex]\( L = 9x^2y^4 \)[/tex].

2. Calculate [tex]\( W = (2x^2)^4 \)[/tex]:

- Raise each component inside the parentheses to the power of 4.
- [tex]\( 2^4 = 16 \)[/tex].
- [tex]\( (x^2)^4 = x^{2 \times 4} = x^8 \)[/tex].

So, [tex]\( W = 16x^8 \)[/tex].

3. Find the area [tex]\( A = L \times W \)[/tex]:

- Multiply the coefficients: [tex]\( 9 \times 16 = 144 \)[/tex].
- Combine the terms involving [tex]\( x \)[/tex]: [tex]\( x^2 \times x^8 = x^{2 + 8} = x^{10} \)[/tex].
- Combine the terms involving [tex]\( y \)[/tex]: [tex]\( y^4 \times 1 = y^4 \)[/tex] (since there’s no [tex]\( y \)[/tex] in [tex]\( W \)[/tex], it’s just [tex]\( y^4 \)[/tex]).

Therefore, the area is [tex]\( A = 144x^{10}y^4 \)[/tex].

The calculated area is expressed as [tex]\( 144x^{10}y^4 \)[/tex], with the coefficient 144, the power of [tex]\( x \)[/tex] as 10, and the power of [tex]\( y \)[/tex] as 4.

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