Question

2x-1 is a factor of f(x) =12x^3+kx^3+mx-3. as well when f(x) is divided by x+2 the remainder is -65. determine the value of m and k.

Answer 1

if 2x-1 is a factor, then 2x-1 = 0 => x = 1/2 use synthetic division to find an equation in k and m.

then if x+2 gives a remainder of -65 => when x = -2, f(-2) = -65. you can use synthetic division again to get another equation in k and m.

now you have a system and can solve for k and m.

Answer 2

use synthetic division:

2x-1 ---> x = 1/2 , remainder = 0

x+2 --> x = -2 , remainder = -65

1/2 | 12 k m -3
6 1/2(k+6) m/2 +(k+6)/4
-----------------------------------------------------
12 k+6 m+(k+6)/2 0

---> m/2 + (k+6)/4 = 3 ----> m = (6-k)/2

-2 | 12 k m -3
-24 48-2k 4k -2m -96
------------------------------------------------
12 k-24 m+48-2k -65

-----> 4k -2m -99 = -65
---> 2k - m = 17

Substitute:
2k - (6-k)/2 = 17
5k - 6 = 34
k = 8 ----> m = -1

Answer 3

wouldn't we have to divide m/2+(k+6)/4 by 2 because we just divided by 1/2 and not 2

Answer 4

we divided by x-1/2 not 2

Answer 5

well, i've never learnt synthetic division as its out of my syllabus

but there's a simpler method to simply sub in the x values

as 2x-1 is a factor,
2x=1
x=1/2
you got that, yes.

so simply make replace all the x in your equation as 1/2
factor means your equation will be = 0

therefore
12(1/2)^3+k(1/2)^3+m(1/2)-3=0
simplify this first

now, for two unknowns, you need two equations
do the same thing for x+2, where x is now = -2
and instead of 0, use -65

substitute
and solve

Answer 6

sorry i dont follow....m/2 +(k+6)/4 is just the result of using synthetic division

Answer 7

but we are divding the polynomial by x+1/2 which comes out of 2(x+1/2), so don't we also divide by 2 that is factored out?

Answer 8

yeah @Auxuris that is prob easier

Answer 9

yeah thanks

Answer 10

ahh i see, well thankfully remainder is 0 so dividing by 2 does not affect the remainder at all

Answer 11

but if the remainder was not zero we would have to divide by 2 right?

Answer 12

yes but only the remainder would be divided by 2..... not the m/2 +(k+6)/4 part
the synthetic div process remains same

Answer 13

so if m/2+(k+6)/4 + (-3) = 5, we would divide 5/2?

Answer 14

yes so you would equal it to 5/2 instead of 5

Answer 15

okay, makes sense

Answer 16

thanks for your help!

Answer 17

yw