Question

can anyone help me with this question
Use the fact that f(-2)=0 to factor f(x)=x^3-3x^2-16x-12

Answer 1

well, if f(-2)=0, x+2 has to be a factor right?

Answer 2

oh yeah i never realized that thanks for pointing that out for me

Answer 3

the other factor has one degree lower than the given cuz x+2 is first order

Answer 4

which means the factor is (x+2)(ax^2+bx+c) and a is not 0

Answer 5

now set two equal to each other, expand LHS and compare coefficients to determine a,b,c if the secondary polynomial has real roots, rewrite the polynomial as (x+x1)(x+x2) where x1and x2 are the roots

Answer 6

im lost now

Answer 7

Ok lets work this out, you undertand that x+2 is one of the factors right

Answer 8

yes

Answer 9

and the other part has the form ax^2+bx+c,right?

Answer 10

where did you get the ax and the bx and the c

Answer 11

Do you mean how did I come up with this form?

Answer 12

yes

Answer 13

well, 5^3 can be factored into 5^2 and 5, right

Answer 14

same thing here, we pull a x out of x^3,the left part has to be x^2

Answer 15

ok

Answer 16

so we have the answer already, it is (x+2)(ax^2+bx+c)

Answer 17

therefore it has to equal to x^3-3x^2-16x-12

Answer 18

so whats the answer

Answer 19

we are getting there, still a long way to go actually

Answer 20

so we have (x+2)(ax^2+bx+c)=x^3-3x^2-16x-12

Answer 21

therefore ax^3+(2a+b)x^2+(2b+c)x+2c=RHS

Answer 22

now we have a=1,2c=-12,2a+b=-3, so you can solve for a b and c

Answer 23

ok

Answer 24

so tell me what are a b and c

Answer 25

b=-5

Answer 26

ok, good

Answer 27

now set ax^2+bx+c=0, is there any real root?

Answer 28

no

Answer 29

well, yes actually b^2-4ac text remember?

Answer 30

yeah sorry

Answer 31

so (-5)^2-4*-6=49, which is neat, because it is 7^2

Answer 32

now can you solve the equation?

Answer 33

yes

Answer 34

so what is the solution?

Answer 35

isn't 49

Answer 36

I am the quadratic equation what are the roots, there are two, if you haven't realized already

Answer 37

well thanks for the help i got to go now