Question

solve the polynomial equation

15x^3-119x^2-10x+16=0

how do I do this?

Answer 1

equate it

Answer 2

drawing a graph will give you an approximate answer
also as 16 is the last term a factor of 16 might be a root
ie 2, 4 or 8
i'll try on my calculator...

Answer 3

yup - 8 is a root
f(8) = 0
so x-8 is a factor

so divide f(x) by (x - 8) to get a quadratic function to find the other roots

Answer 4

* 8 is a root , not -8

Answer 5

*votes @cwrw238 for best answer* In addition to what he told you already, he is using the rational root theorem. You might want to look into that yourself.

Answer 6

(x-8)(15x^2+x-2) ---> x-8/15x^+x-2 right?

Answer 7

rational root theorem - i must admit i've never heard of it

Answer 8

http://www.mathwords.com/r/rational_root_theorem.htm

For both of you then (-;

Answer 9

ok thanks

yes santanaG - look right

Answer 10

ok then once I get the answer from that, that should be my answer?

Answer 11

oh ok - i must admit - i didn't know that the coefficient of x^3 was involved - though it makes sense that it is!!!!!

Answer 12

yes - looks like theres 3 real roots

Answer 13

rational root theorem - i was using half of it without realizing it !!!!!

Answer 14

hehehe, impressive @cwrw238, well now you have one more neat theorem in your math dictionary (-: The interesting thing is that in almost all the cases I have met so far, it's enough to analyze the constant at the end, I do it the same way as you. Only if I can't find a solution that way I include the coefficient of the leading term.

Answer 15

right - thanks

Answer 16

i got 15=1,3,5.15
16=1,2,4,6,8,16


and then did p/q with all of them... and thats my answer is all of them as p/q

@Spacelimbus and @cwrw238

Answer 17

these are your possible rational zeros @SantanaG, once you have found these, you can 'try and error' and see which one fits your polynomial to make this expression true:

\[p(x_0)=0\] where x can be any number of the rational roots you have discovered.

Answer 18

x=8,-2/5,1/3

Answer 19

you can discover all the rational roots of a polynomial with this method, but keep in mind that there can be irrational solutions too, for example the quadratic equation you obtain by reducing the degree (dividing by x-a) can give you additional solutions.

Answer 20

the remaining roots are:
\[x_2= 0.4 \\x_3=-\frac{1}{3} \]

Answer 21

sorry, I got the minus and plus wrong, here again:

\[x_2=-0.4 \\ x_3= \frac{1}{3} \]