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