Question

The area of a rectangle is given by

10
b
3
-
15
b
2
+
14
b. The width of the same rectangle is given by
b
-
1. Find the length of the rectangle. Remember, the the length times the width of a rectangle is equal to its area. The length of the rectangle is

(If there is a remainder
r
, express it in the form

r
b
-
1. )

Answer 1

The area of the rectangle can be expressed as:

10b^3 - 15b^2 + 14b

We know that the width of the rectangle is b - 1, so we can express the length of the rectangle in terms of b as:

length = area/width

length = (10b^3 - 15b^2 + 14b)/(b - 1)

To simplify this expression, we can use polynomial long division:

10b^2 - 5b - 9

___________________

b - 1 | 10b^3 - 15b^2 + 14b

- (10b^3 - 10b^2)

-------------------

-5b^2 + 14b

- (-5b^2 + 5b)

--------------

9b

- 9

-----

0

Therefore, the length of the rectangle is 10b^2 - 5b - 9. There is no remainder, so r = 0.

Answer: The length of the rectangle is 10b^2 - 5b - 9.