Final answer:
The polynomial representing the perimeter of the rectangular frame with sides labeled 3x + 1 and 5x - 2 is 16x - 2.
Step-by-step explanation:
To write a polynomial that represents the perimeter of a rectangular frame with the given dimensions, we need to sum the lengths of all four sides.
The perimeter (P) of a rectangle is given by twice the width (w) plus twice the length (l), which can be expressed as P = 2w + 2l. Given the dimensions are labeled as 3x + 1 for the width and 5x - 2 for the length, the polynomial for the perimeter is:
P = 2(3x + 1) + 2(5x - 2)
Expanding the parentheses:
P = 2×3x + 2×1 + 2×5x - 2×2
P = 6x + 2 + 10x - 4
Combining like terms:
P = (6x + 10x) + (2 - 4)
P = 16x - 2
So, the polynomial representing the perimeter of the picture frame is 16x - 2.