The polynomial that represents the length is:
h(x) = 11x + 1.
How to find the length?
We know that the area of the top surface is given by:
f(x) = 66x^2 + 17x + 1
And the width of the top surface is:
g(x) = 6x + 1.
We can assume that the length, h(x), is another polynomial of degree 1, so we can write:
h(x) = a*x + b
Now, the area is the product of the width and the length, so we have:
f(x) = g(x)*h(x)
f(x) = (6x + 1)*(ax + b)
= 6ax^2 + (6b)x + (a)x + b
= (6a)x^2 + (6b + a)x + b
Now let's compare this with f(x), correspondent coefficients must have the same value.
66x^2 + 17x + 1 = (6a)x^2 + (6b + a)x + b
Then we have:
66 = 6a
17 = 6b + a
1 = b
From the first equation we have:
66/6 = a = 11
And from the third:
b = 1
Then the length is:
h(x) = 11x + 1.
If you want to learn more about polynomials, you can read:
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