Question

Given the polynomial f(x)=x^3+ax^2+bx+12 gives a remainder of 32 when divided by (x-5) and a remainder of 20 when divided by (x+1), solve the equation of f(x)=0

Answer 1

12

Answer 2

if we put x=0 ..den all d term vanishes and we get f(0)=12

Answer 3

its not asking about x = 0 though, its asking about f(x) = 0

Answer 4

i thought what you put at first, but i was mistaken.

Answer 5

ohh srry ..

Answer 6

basically you know that f(5) = 32, and f(-1) = 20, you can create a system of equations in a and b. You have 2 equations, 2 unknowns. Then once you figure out a and b, put the values in, and find the zeros of the polynomial by factoring.

Answer 7

@Joe, yeah ;; I'm at a lost. I think i should do
\[R_{1}\]=32 fx
...I really don't know.

Answer 8

\[f(5) = 32 \iff (5)^3+a(5)^2+b(5)+12 = 32 \iff 25a+5b = -105\]

Answer 9

how do u knw f(5)=32??

Answer 10

f(-1) = 20 gives a-b = 9

Answer 11

(x-b)
f(b)
um.. right?

Answer 12

yes

Answer 13

when you divide f(x) by (x-5) you will get a quotient and a remainder. that looks like this:
\[\frac{f(x)}{(x-5)} = q(x)+\frac{r(x)}{(x-5)}\]
multiplying by x-5 we get:
\[f(x) = (x-5)q(x)+r(x)\]
If you plug in x = 5 you get:
\[f(5) = (5-5)q(5)+r(5) = r(5) = 32\]
since they told us the remainder was 32

Answer 14

i believe you should get a is -2, and b is -11 from that system.

so the polynomial is now:
\[x^3-2x^2-11x+12\]
Now we just need to find the roots.

Answer 15

1 is a root.
by inspection.

Answer 16

OMG thanks I get it now
25a + 5b = -105
-(25a - 25b= 225)
-------------
30b= -330
b=-330:30
b=-11

is that right...?
I'm a bit slow sorry

Answer 17

im studying for a final atm, so im sry if it seems like im not fullying explaining things. I would be glad to go over again after my test >.< lol

Answer 18

and yes thats right, b is -11, now you just need a.

Answer 19

Joe I love you ;w; good luck for your finals!!

Answer 20

a-(-11)=9
a=9-11
a=-2

Answer 21

right :)
alright im off, good luck with your problem. i'll check it after i get out.
thanks for the good luck :)

Answer 22

You're a genius, thank you~