Given that f(x) = 2x³-7x²+7ax+ 16 is divisible by x-a, find (i) the value of the constant a
(ii) the remainder when f(x) is divided by 2x+1.

a = - 2, remainder = 21

Step-by-step explanation:

The Remainder theorem states that if f(x) is divided by (x - a) the remainder is f(a)

Since f(x) is divisible by (x - a) then remainder is zero , then

f(a) = 2a³ - 7a² + 7a² + 16 = 0 , that is

2a³ + 16 = 0 ( subtract 16 from both sides )

2a³ = - 16 ( divide both sides by 2 )

a³ = - 8 ( take the cube root of both sides )

a = = - 2

Then

f(x) = 2x³ - 7x² - 14x + 16

Evaluate f(- ) for remainder on division by (2x + 1)

f(- ) = 2(- )³ - 7(- )² - 14(- ) + 16

        = 2(- ) - 7( ) + 7 + 16

        = - - + 23

        = - + 23

       = - 2 + 23

       = 21

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional