Polynomial Practice worksheet featuring a bedroom diagram with a rug, asking students to write algebraic expressions for various measurements.

Let’s solve this step by step.

We are told:
- A rug is placed in the middle of a bedroom.
- The length of the rug is two feet more than the width.
- Let’s use variables to represent dimensions.

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Step 1: Define variables

Let:
- \( w \) = width of the bedroom (in feet)
- Since the rug is centered and there’s space around it, we need to figure out how much space is on each side.

Looking at the diagram description (even though we can’t see it, the problem implies):
- There is 2 feet of floor visible on each side of the rug — left/right and top/bottom.
- So, if the bedroom has width \( w \), then the rug’s width = \( w - 4 \) (because 2 ft on left + 2 ft on right = 4 ft total subtracted).
- Similarly, if the bedroom has length \( l \), then the rug’s length = \( l - 4 \).

But wait — the problem says:
> “The length of the rug is two feet more than the width.”

That refers to the rug’s own dimensions, not the room’s.

So let’s define:

Let \( x \) = width of the rug (in feet)
Then, length of the rug = \( x + 2 \)

Now, since the rug is centered with 2 feet of space on all sides:

→ Width of bedroom = rug width + 2 (left) + 2 (right) = \( x + 4 \)
→ Length of bedroom = rug length + 2 (top) + 2 (bottom) = \( (x + 2) + 4 = x + 6 \)

Wait — that might be overcomplicating. Let’s read the questions again.

Actually, looking at the worksheet structure:

It asks for expressions in terms of w (width of bedroom) and l (length of bedroom). So probably:

Let:
- \( w \) = width of bedroom
- \( l \) = length of bedroom

From the diagram (implied): rug is centered with 2 ft border on all sides → so:

Rug width = \( w - 4 \)
Rug length = \( l - 4 \)

And we’re told:
> “The length of the rug is two feet more than the width.”
→ So:
\( l - 4 = (w - 4) + 2 \)
Simplify:
\( l - 4 = w - 2 \)
→ \( l = w + 2 \)

Interesting! So the bedroom’s length is 2 feet more than its width? Wait — no:

Wait:
Rug length = rug width + 2
→ \( l - 4 = (w - 4) + 2 \)
→ \( l - 4 = w - 2 \)
→ \( l = w + 2 \)

Yes — so the bedroom’s length is 2 feet more than its width.

But actually, let’s double-check:

If rug width = w - 4
Rug length = l - 4
And rug length = rug width + 2
Then:
l - 4 = (w - 4) + 2
l - 4 = w - 2
l = w + 2

✔ Correct.

So now we can answer the questions.

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## Question 1: Write an expression in terms of w for the length of the bedroom.

We just found:
Length of bedroom = w + 2

Answer: \( w + 2 \)

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## Question 2: Write an expression in terms of w for the width of the rug.

Rug width = bedroom width minus 4 (2 on each side)
→ \( w - 4 \)

Answer: \( w - 4 \)

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## Question 3: Write an expression in terms of w for the area of the bedroom.

Area = length × width = \( (w + 2) \times w = w(w + 2) \)

Or expanded: \( w^2 + 2w \)

Either form is acceptable, but since it says “expression”, factored or expanded is fine. We’ll write both possibilities, but likely they want simplified.

Actually, let’s keep as \( w(w + 2) \) unless specified otherwise. But often in these problems, they accept either. Let’s go with expanded for clarity.

Answer: \( w^2 + 2w \)

Wait — actually, let’s check question 4 too.

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## Question 4: Write an expression in terms of w for the area covered by the rug.

Rug width = \( w - 4 \)
Rug length = ? From earlier: rug length = rug width + 2 = \( (w - 4) + 2 = w - 2 \)

Alternatively, from bedroom length: l = w + 2 → rug length = l - 4 = (w + 2) - 4 = w - 2 ✔ same.

So rug area = width × length = \( (w - 4)(w - 2) \)

Expand:
= \( w^2 - 2w - 4w + 8 = w^2 - 6w + 8 \)

Answer: \( w^2 - 6w + 8 \)

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## Question 5: Write an expression in terms of w for the area of the floor NOT covered by the rug.

That’s bedroom area minus rug area.

Bedroom area: \( w^2 + 2w \)
Rug area: \( w^2 - 6w + 8 \)

Subtract:
\( (w^2 + 2w) - (w^2 - 6w + 8) = w^2 + 2w - w^2 + 6w - 8 = 8w - 8 \)

Answer: \( 8w - 8 \)

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## Question 6: If the uncovered floor measured 296 square feet, find w.

Set equation:
Uncovered area = 8w - 8 = 296

Solve:
8w - 8 = 296
Add 8 to both sides:
8w = 304
Divide by 8:
w = 38

So width of bedroom = 38 feet

Then length of bedroom = w + 2 = 40 feet

Check:

Rug width = 38 - 4 = 34
Rug length = 40 - 4 = 36 → which is 2 more than 34 ✔

Bedroom area = 38 × 40 = 1520
Rug area = 34 × 36 = let’s compute:
34×36 = (30+4)(30+6) = 30² + 30×6 + 4×30 + 4×6 = 900 + 180 + 120 + 24 = 1224
Uncovered = 1520 - 1224 = 296 ✔ correct.

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## Question 7: What are the dimensions of the bedroom?

Width = 38 ft, Length = 40 ft

Answer: 38 feet by 40 feet

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## Question 8: What are the dimensions of the rug?

Width = 34 ft, Length = 36 ft

Answer: 34 feet by 36 feet

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Now, let’s fill in the blanks as per the worksheet order.

Note: The worksheet lists:

1. Expression for length of bedroom in terms of w → \( w + 2 \)

2. Expression for width of rug in terms of w → \( w - 4 \)

3. Expression for area of bedroom in terms of w → \( w(w + 2) \) or \( w^2 + 2w \)

4. Expression for area of rug in terms of w → \( (w - 4)(w - 2) \) or \( w^2 - 6w + 8 \)

5. Expression for uncovered area in terms of w → \( 8w - 8 \)

6. Given uncovered area = 296, find w → w = 38

7. Dimensions of bedroom → 38 ft by 40 ft

8. Dimensions of rug → 34 ft by 36 ft

All checks out.

Final Answer:

1. \( w + 2 \)
2. \( w - 4 \)
3. \( w^2 + 2w \)
4. \( w^2 - 6w + 8 \)
5. \( 8w - 8 \)
6. \( w = 38 \)
7. 38 feet by 40 feet
8. 34 feet by 36 feet