Certainly! Let's write a detailed, step-by-step solution for this problem.
We are given that the length of a picture frame is 12 inches less than twice the width of the frame. We need to find a polynomial expression for the area of the picture frame in terms of the width, \(\( w \)\).
1. Define Variables:
- Let \(\( w \)\) be the width of the picture frame.
- According to the problem, the length \(\( L \)\) is 12 inches less than twice the width.
2. Express the Length in Terms of \(\( w \)\):
- Twice the width is \(\( 2w \)\).
- The length is then 12 inches less than this, which gives us:
\(\[
L = 2w - 12
\]\)
3. Write the Expression for the Area:
- The area \(\( A \)\) of a rectangle is given by the product of its length and width.
\(\[
A = \text{Length} \times \text{Width}
\]\)
- Substituting the expressions for length and width:
\(\[
A = (2w - 12) \times w
\]\)
4. Simplify the Expression:
- Distribute \(\( w \)\) in the product:
\(\[
A = (2w - 12)w
\]\)
\(\[
A = 2w^2 - 12w
\]\)
5. Write the Final Polynomial Expression in Standard Form:
- The polynomial expression for the area, in terms of \(\( w \)\), is:
\(\[
A = 2w^2 - 12w
\]\)
Therefore, the polynomial expression for the area of the picture frame, given the width \(\( w \)\), is:
\(\[
A = 2w^2 - 12w
\]\)
