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The area of a rectangle is represented by the function x3 − 2x2 − 40x − 64. The width of the rectangle is x + 4. Find the expression representing the length of the rectangle.

A. x2 − 8x − 20
B. x2 + 12x + 20
C. x2 − 6x − 16
D. x2 + 10x + 16

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Answer:

Step-by-step explanation:

If the area is

[tex]x^3-2x^2-40x-64[/tex]

and area = length * width and our width is given as x + 4, then the length is found by dividing the area by the width.  You could do that using long division, but it's easier using synthetic division.

-4 |   1   -2   -40   -64

    ______________

       1

This is how you start.  The -4 inside the box comes from the factor you are dividing by.  If x + 4 = 0, then x = -4.  The numbers after are the coefficients from each descending power of x.  Multiply the -4 by the 1 and put that product up under the -2 and add to get:

-4 |   1   -2   -40   -64

            -4

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

       1     -6

Now multiply -4 by -6 and put that product up under the -40 and add:

-4 |  1    -2  -40   -64

           -4   24

   -------------------------

      1    -6    -16

Multiplying -4 by -16 gives you 64 so when you add you get a remainder of 0.

The numbers under the line give you the depressed polynomial

[tex]x^2-6x-16[/tex]

That gives you the expression for the length.  That's C.        

The expression representing the length of the rectangle is option C;  x^2 − 6x − 16.

What is the area of the rectangle?

The area of the rectangle is the product of the length and width of a given rectangle.

The area of the rectangle = length × Width

The area of a rectangle is represented by the function;

x^3 − 2x^2 − 40x − 64.

The width is given as x + 4,

We could do that using long division, but it's easier using synthetic division.

-4 |   1   -2   -40   -64

   ______________

      1

The -4 inside the box comes from the factor you are dividing by.

If x + 4 = 0, then x = -4.  

Now, Multiply the -4 by the 1 and put that product up under the -2 and add to get:

-4 |   1   -2   -40   -64

           -4

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

      1     -6

Now, multiply -4 by -6 and put that product up under the -40 and add:

-4 |  1    -2  -40   -64

          -4   24

  -------------------------

     1    -6    -16

Again, Multiplying -4 by -16 gives you 64 so when you add you get a remainder of 0.

The numbers under the line give the depressed polynomial;

x^2 − 6x − 16

Thus, the expression representing the length of the rectangle is option C;  x^2 − 6x − 16.

Learn more about the area;

https://brainly.com/question/1658516

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