The shipping container is a rectangular prism. Write a polynomial that represents
the volume of the container.

The volume of a rectangular prism is simply the product of its three dimensions: in your case, the volume of the prism is, given  

x

,

(

x

+

6

)

(

x

−

2

)

(

x

−

1

)

.

A polynomial is a sum (with some coefficients) of powers of  

x

, so, if we expand the product just written, we have

(

(

x

+

6

)

(

x

−

2

)

)

(

x

−

1

)

=

(

x

2

−

2

x

+

6

x

−

12

)

(

x

−

1

)

=

(

x

2

+

4

x

−

12

)

(

x

−

1

)

=

x

3

+

4

x

2

−

12

x

−

x

2

−

4

x

+

12

=

x

3

+

3

x

2

−

16

x

+

12

Which is a polynomial, and expresses the volume of the prism

Step-by-step explanation:

The volume of a rectangular prism is simply the product of its three dimensions: in your case, the volume of the prism is, given  

x

,

(

x

+

6

)

(

x

−

2

)

(

x

−

1

)

.

A polynomial is a sum (with some coefficients) of powers of  

x

, so, if we expand the product just written, we have

(

(

x

+

6

)

(

x

−

2

)

)

(

x

−

1

)

=

(

x

2

−

2

x

+

6

x

−

12

)

(

x

−

1

)

=

(

x

2

+

4

x

−

12

)

(

x

−

1

)

=

x

3

+

4

x

2

−

12

x

−

x

2

−

4

x

+

12

=

x

3

+

3

x

2

−

16

x

+

12

Which is a polynomial, and expresses the volume of the prism

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1

1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.