Question

find the roots for f(x) = x4 + 21x2 – 100.

Answer 1

@kc_kennylau do u know how to do this?

Answer 2

first, setting f(x) = 0, in other words
x^4 + 21x^2 - 100 = 0
to make be easier, let k = a^2. so the equation equals
k^2 + 21k - 100 = 0
solve for k, first. like to get the roots of quadratic

Answer 3

how do i solve for k ? @RadEn

Answer 4

k^2 + 21k - 100 = 0
the equation above can be factored :
(k + 25)(k - 4) = 0
take each factor equals zero, then solve for k

Answer 5

k+25=0
k = .... ?

also
k-4=0
k= .... ?

Answer 6

-25 & 4

Answer 7

that's correct!

now return k = a^2
so, we have 2 cases :

a^2 = -25 and

a^2 = 4

solve for a

Answer 8

would it be 5, -5 and 2, -2?

Answer 9

2 and -2, these roots are correct. but not yet for 5 and -5

a^2 = -25
a = +- sqrt(-25)
a = +- sqrt(25 * -1)
a = +- sqrt(25) sqrt(-1)
a = +- 5 sqrt(-1)
a = +- 5i

i assumed the complex roots is allowed here

Answer 10

so it would be -5i and 5i?

Answer 11

yes, all the roots exactly :
{-5i, 5i, -2, 2}

Answer 12

ok thank you so much!

Answer 13

you're welcome

Answer 14

could you help me with one more problem? @RadEn

Answer 15

i'll try, what is it ?

Answer 16

find the roots for f(x) = x3 - 5x2 – 25x + 125.

Answer 17

x^3 - 5x^2 – 25x + 125 = 0

first, can you find the factors of 125 ?

Answer 18

1,5,25,125,

Answer 19

sometimes we need the negative sign,
125 = {+-1, +-5, +-25, +-125}

Answer 20

oh ok, so what do we do next?

Answer 21

one of those must be as the root of f(x), just trials which they are
let it is 5. recheck if this be correct :
f(x) = x^3 - 5x^2 – 25x + 125
f(5) = (5)^3 - 5(5)^2 – 25(5) + 125
f(5) = 125 - 125 - 125 + 125
f(5) = 0

because f(5) = 0, so 5 is the root of f(x)

Answer 22

ok so next we need to find 2 more roots right? because the exponent is 3 for the x

Answer 23

yes, that's right. actualy we can use the long division or horner to get the others

Answer 24

and how would we do that? im sorry im making u do all the work im just trying to learn this

Answer 25

nvm, if i can why i didnt help the members here :)

see how the diagram horner works :