CPhill

Remember GM....if x - 4 is a factor then 4 is a root

This is where the Remainder Theorem comes in handy

So....if  4 is a root....then  putting 4 into the function will return 0....so....

(4)^2 +  k(4) - 52 = 0   simplify

16  + 4k   - 52 = 0

-36 + 4k = 0      add 36 to both sides

4k =  36           divide both sides by 4

k = 9

So....we have    x^2 + 9x - 52

Notice that this factors  as  ( x - 4) (x + 13)

Note that (x - 4) is a factor

g(x) =  -2x ,  x ≥3

Let's just pick two x values ≥ 3   and stick them into the function

When x = 3,  y = -6.....so....(3, -6) is on the graph

When x = 4, y = -8.....so.....(4, -8) is on the graph

Plot  (3, -6)

Then...plot (4, -8).....draw a ray startting at (3, 6) and extend it through (4, -8)

g(x) = -(1/3)X - 5,  x ≤3

Again..let's pick x = 3   and put it into the function....and we have (-1/3)(3) - 5 = -1 - 5  = -6

So....the point (3, -6) is on the graph

Now.....pick anoher x value < 3.......0 is good

So   (-1/3)(0) - 5 =  -5

So (0, -5) is on this graph

Draw a ray from (-3, 6) through (0, -5)

The graph will look like this :

Of course...the lines (rays) will have "arrows" at their ends  

Multiples of 7 that exceed  what ???

First one.....we write  the terms in descending powers of   m......so we have

4m^3 - 9 m^2 - m + 25     (cubic polynomial)

Second one  ...... same thing ....descending powers of k  ....so we have

(3/4)k^5  - (1/2)k^3       (quintic polynomial)

Third one.....this is a polynomial  in two variables

The form is     ax^2 + bxy + cy^2 + dx + ey + f 

So we have

-x ^2   -3xy  + 14y^2    (second power polynomial)

Last one.....nothing to be done.....there's no other way to write 9x  ( first degree polynomial )

5.6^(-5) * 3.4^(-7) * 5.6^(3) * 3.4^(-4)  =

 5.6^(-5) * 5.6^(3) * 3.4^(-7) * 3.4^(-4)        { we are using the rule a^m * a^n = a^(m + n)   }

5.6^(-2) * 3.4^(-11)

Here's a graph of the first one :   https://www.desmos.com/calculator/oqouuljrdk

Second one :  The graph at the top left is the correct one

Third one :   f(x)  =  10cos  [ (pi/2) x ] + 10    is correct

Last one :  f(x) = 9 sin ( 6pi x) + 3    is correct

OK.....glad to help....!!!

LOL!!!!

No, THX.....even looking at pics of snow starts me shivering  !!!!

Are you in CO, EP????