CPhill

What Melody said  !!!

f(4)   is equal to the remainder, i.e., 14

l 2x  - 8  l  + 3  = 5    subtract  3 from both sides

l 2x - 8 l   = 2      

This actually says that we have these two equations

(2x - 8)  = 2  or   - (2x - 8)  = 2

So...answer "K"

To see the reason for this suppose we have   l a l  = 9....note that two values of  "a"  make this true...either  a = 9  or  a = -9

No..... if  f(x)  is divided by   x - 3....we are actually testing whether  "3"  is a root  [ it isn't because we get a remainder]

So f(3)  = 15, not  f(-3)

1)

x^2 - 3x  = 18     subtract  18 from both sides

x^2  - 3x - 18  = 0    factor the left side

(x - 6) ( x + 3)   = 0

Set both factors to 0  and solve for x and we get that   x  = 6   or  x = -3

2) 

x^2   - 3x +  ___   =  18  +   ___

Take 1/2 of 3  = 3/2...square this  = 9/4...add to both sides

x^2  - 3x + 9/4  =  18 + 9/4        factor the left side  and simplify the right

(x - 3/2)^2  = 72/4 + 9/4

(x - 3/2)^2  =  81/4         take both roots 

x - 3/2  = ±√ [ 81/4]    take the square root on the right

x - 3/2 = ± 9/2     add 3/2 to both sides

x = 3/2 ±9/2   which means that  x  = 3/2 + 9/2  = 12/2  = 6   or   x = 3/2 - 9/2  = -6/2  = -3

1)

4x^2 + 16x  = 48      subtract 48 from both sides

4x^2 + 16x  - 48  = 0      factor out the 4

4(x^2 + 4x - 12)  = 0      divide away the 4

x^2 + 4x - 12   = 0        factor this

(x + 6) ( x - 2)   = 0

Set both factors to 0  and solve for x and we get that   x  = -6   or  x = 2

2)

x^2 + 7x  = -6    add 6 to both sides

x^2 + 7x + 6  = 0    factor the left side 

(x + 6) ( x + 1)   = 0     

Set both factors to 0  and solve for x and we get that   x  = -6   or  x = -1

Let the integers be   x, x + 1 and x + 2

So we have

4x  =  x+ 2 + 13    simplify

4x  = x + 15    subtract x from both sides

3x  = 15     divide both sides by 3

x  = 5

So...the integers are 5, 6, 7

D. 36

Let the portion Jenny can eat in one hour  = 1/x

Let the portion  that  Penny can eat in  one hour = 1/y

Let the portion that Lenny can eat in one hour = 1/z

And we have this system

2/x + 2 / y  = 1      (1)

(5/3)/y + (5/3)/z =  1      (2)      

1/x + 1/y + 1/z  = 1   (3)

Multilpy (2)  by  (3/5)  and we have

1/y + 1/z  = 3/5     (4)

Put (4) into (3) and we have

1/x + 3/5  = 1     subtract   3/5 from  both sides

1/x = 2/5  ⇒   x  = 5/2  = 2.5 hrs = time for Jenny to eat the whole gallon by herself

Multiply the first equation  by  -2  and we have

-2x  - 2y  - 6z  = -20     

-4x  + 2y + 5z  = 7          add these  and we get

-6x   - z    =  -13     (1)

kx  +   z   =      3

Multiply (1)  by - 1

6x + z  =  13

kx + z  = 3

This  system will have no solution  when k  =  6