Use diagrams or the description of equivalent ratios to show that the ratios 2: 3, 4: 6, and 8: 12 are equivalent.

4:6 and 8:12 are equivalent ratios to 2:3 as you can see in the diagram below

4:6

_2_  _2_  

_2_  _2_  _2_  

8:12

_4_ _4_

_4_ _4_ _4_

Step-by-step explanation:

The diagrams of equivalent ratios represent the initial relation, in the image you can see each value is in a box so as the initial relation is 2:3 we have two boxes in the first row and three on the second. Now we have to find a number that added the times of the box, give us the other ratios 4:6 and 8:12

4:6

_2_  _2_  = 4

_2_  _2_  _2_    = 6

As 4 is 2 times 2 and 6 is 3 times 2, 4:6 is an equivalent ratio to 2:3 where 2 is the constant.

8:12

_4_ _4_ = 8

_4_ _4_ _4_ =12

As 8 is 2 times 4 and 6 is 3 times 4, 8:12 is an equivalent ratio to 2:3 where 4 is the constant.

To calculate the constant you may divide each of the corresponding values for example 4 divided by 2 is 2 and 6 divided by 3 is 2 then the constant for 4:6 is 2.

In the case of 8:12, 8 divided by 2 is 4 and 12 divided by 6 is 4 then the constan for 8:12 is 4

(x3 + 6x2 + 2x - 6) • (x - 1)

step-by-step explanation:

polynomial long division :

3.2     polynomial long division

dividing :   x4+5x3-4x2-8x+6

                              ("dividend")

by         :     x-1     ("divisor")

dividend     x4   +   5x3   -   4x2   -   8x   +   6

- divisor   * x3     x4   -   x3            

remainder         6x3   -   4x2   -   8x   +   6

- divisor   * 6x2         6x3   -   6x2        

remainder             2x2   -   8x   +   6

- divisor   * 2x1             2x2   -   2x    

remainder               -   6x   +   6

- divisor   * -6x0               -   6x   +   6

remainder                     0

quotient :   x3+6x2+2x-6   remainder:   0