CPhill

Q5

Time * Rate  =  Distance  ....so.....

Mary drives between  4(40)    = 160 miles   and less than 4(60)  =  240  miles

Since Marty  travels 50  miles more, he travels between  210  and   290  miles

Q4

( x^4 * y^7) ^(3/4)  =

 ⁴√  [ (x^4)^3 *  (y^7)^3 ] =

⁴√  [x^12 * y^21 ] =

x^3 * y^5  ⁴√  y     ⇒   Third answer

Q3

We have the following system

3x + 2y ≥ 300

.75x + .50y ≤ 100

x ≤ 80

Look at the graph, here :  https://www.desmos.com/calculator/welwdepye5

Four of the given points lie in, or on the boundary of, the feasible region

The only one that doesn't  is ( 40, 80 )  [ the only non-feasible point ]

Q2...to find the correct solution...pick a point in the shaded region...I'll pick   (x, y)  =   ( 0,3)

Note that this point satisfies all three equations in the first answer....so this answer is correct

Q1

2√32 + 5√20 =

2 *√16 * √2  +  5 * √4 * √5  =

2* 4 * √2   +  5*2 * √5  =

8√2  + 10√5

Second one

x^4 - x^3 -7x^2 + x + 6

Since 3 is a zero, ( x - 3)  is one linear factor...we can use synthetic division to find the remaining polynomial

3 [  1     - 1       - 7      1        6  ]

               3        6      -3      - 6

     _____________________

     1       2        -1     -2         0

The remaining polynomial is

x^3  + 2x^2  - x  - 2       we can factor this as

x^2 (x + 2)  - 1 (x + 2)   =

(x + 2) ( x^2 - 1)  =

(x + 2) ( x + 1) ( x - 1)

So.......the factored form of the polynomial is  ( x - 3)(x + 1) (x - 1) ( x + 2)

First one

1x^4  - 4x^3 -22x^2 + 4x + 21

First of all...if we can add the coefficients and the constant at the end and get 0, then  1 is a root ...and this is true

So...we can use some sythetic division to find the other  roots

1 [  1     - 4      - 22       4       21]

               1        -3       -25    -21

     _______________________

      1      -3       - 25     -21     0

The remaining polynomial is   x^3  - 3x^2  - 25x  - 21

By the Factor Theorem   the remaining  possible zeroes  are ±1, ±3, ±7  or ±21

We can test these values in the polynomial....those that result in  "0"  means that that integer is a root

-1 is a root

-3 is a root 

7 is a root

We can stop here....a 4th power polynomial can't have more than  4 roots

So...the roots are   x = { -3, -1, 1 , 7 }

Here you go .....here's the way I solved it...

https://web2.0calc.com/questions/i-ve-tried-solving-this-and-every-time-i-get-a-different

I've got to go now,NSS....but...I'll come back later and show you the general proceedure.....!!!

Well....if I keyed everything in correctly....

Variance  = 2056.52

Std Dev   = 45.35

EDIT to correct an earlier error....