Question

find all the zeroes and write a linear factorization of the function 5x^3-24^2+x+12

Answer 1

familiar wid rational root theorem ?

Answer 2

no as much as I would like to be.

Answer 3

its okay, u heard of it during ur class right ?

Answer 4

Yes. But I didn't quite understand it.

Answer 5

rational root theorem says this :-

```
rational roots will be of form : p/q
p = factors of constant term
q = factors of leading coefficient
```

Answer 6

here ,
constant term = 12
leading coefficient = 5

Answer 7

wat are the factors of 12 ?
wat are the factors of 5 ?

Answer 8

1, 2, 3, 4,5,6,12

Answer 9

Yes, but separate them.
just list factors of 12 in one row
and list factors of 5 in another row

Answer 10

1,2,3,4,6,12/1,5 Like that?

Answer 11

yeah looks good.

start testing each of them one by one

Answer 12

you can do that,
but there is a quicker way.

u knw synthetic division right ?

Answer 13

Yes.

Answer 14

wonderful ! then you have all the weapons !!
use it and see if we get a 0 for any of them

Answer 15

lets test and see if 1/1 = 1 is a zero :-

1 | 5 -24 1 12
| 5 19 20
-------------------------
5 -19 20 42

Answer 16

1 is not a 0,
we're not lucky yet.

Answer 17

we test it on both the negative and positive factors right?

Answer 18

by some magic.... 4/5 looks like our guy.
lets test it and see if we get a 0

Answer 19

yes !
more donkey work...
lets finish positive stuff first, before turningto negatives

Answer 20

could u test and see if 4/5 gives us a 0 ? :)

Answer 21

I think I did it wrong... I didn't get a 0

Answer 22

okay let me do this.

Answer 23

4/5 | 5 -24 1 12
| 4 -16 -12
-------------------------
5 -20 -15 0

we get a 0 !!
so 4/5 is one of the zero.

Answer 24

and the depressed equation is :

5x^2 -20x - 15

Answer 25

use quadratic formula and find other two zeroes

Answer 26

5x^2 -20x - 15 = 0

x^2 - 4x - 3 = 0

Answer 27

apply quadratic formula on above.
(ive just cancelled 5 thru out)

Answer 28

A=1 B=-4 and C=-3?

Answer 29

-4plus or minus square root of 28 divided by 2 ?

Answer 30

Yes !
it simplifies further :o

Answer 31

\(\large \frac{--4\pm \sqrt{28}}{2}\)

\(\large \frac{4\pm \sqrt{4 \times 7}}{2}\)

\(\large \frac{4\pm 2\sqrt{7}}{2}\)

Answer 32

cancel 2

Answer 33

4 plus or minus squareroot of 7?

Answer 34

\(\large \frac{4\pm 2\sqrt{7}}{2}\)

\(\large \frac{2(2\pm \sqrt{7})}{2}\)

\(\large 2 \pm \sqrt{7}\)

Answer 35

Ah I see it now!

Answer 36

so the 3 zeroes are : \(\frac {4}{5}, ~~~ 2 + \sqrt{7}, ~~~~ 2 - \sqrt{7} \)

Answer 37

can u write the linear factorization now ?

Answer 38

thats just a fancy term u knw,
its simple actually, we're done almost :)

Answer 39

linear factorization of given function is just :
(x - zero)(x-zero).....

Answer 40

so the linear factorization of our function is :-

\(\large (x-\frac {4}{5})(x-(2 + \sqrt{7}))(x-( 2 - \sqrt{7})) \)

Answer 41

see if that makes some sense

Answer 42

Okay now I understand.
Thank you!

Answer 43

glad to hear :) yw!