hey, can somebody please help me? i have a HUGE text tomorrow!
how can i find the roots of 3x^4+11x^3+14x^2+7x+1=0? i think this particular equation requires the quadratic formula also along the way

all help is very much appreciated!! thank you!!!

Question

hey, can somebody please help me? i have a HUGE text tomorrow!
how can i find the roots of 3x^4+11x^3+14x^2+7x+1=0? i think this particular equation requires the quadratic formula also along the way

all help is very much appreciated!! thank you!!!

anonymous · Answer

First thing to do is to get the possible roots of the equation. To wit $$3x^4+11x^3+14x^2+7x+1=0$$Get the factors of the coefficient of the term of highest degree (here, $x^4$). The factors of 3 are $\pm 1 \text{ and } \pm3$. 
Now, get the factors of the constant term, (here, 1). The factors of 1 are $\pm{1}$.

To put it altogether:

$p$: (for constant term) $\pm{1}$
$q$: (for the $x^4$ term) $\pm{1}, \pm{3}$

Therefore:

$\frac{p}{q}$: (possible roots of the equation) $\pm{1}, \pm{\frac{1}{3}}$.

Now use synthetic division to test each possible root.

anonymous · Answer

thank you so much @Jnlucero!! but i thought we had to use the quadratic formula in this one?

anonymous · Answer

Nope, you have to do synthetic division first to find some of the roots. When the reduced equation becomes quadratic, that's the time we have to use either factoring or quadratic formula.

anonymous · Answer

can you do me a huge favor and do this one so i can see an example?

anonymous · Answer

@Jnlucero

anonymous · Answer

sure. can you wait for a while?

anonymous · Answer

yes, absolutely, thank you so much!!

anonymous · Answer

_-1_|   3   11    14     7      1        try -1
__________-3___-8___-6__-1
           3    8      6      1      0

Since the last number is zero, then -1 is a root. Now consider only the first four numbers.

_-1_|  3     8      6      1               try -1 again (yes, it can repeat many times)
__________-3___-5___-1
          3     5      1      0

Again, the last number zeroes, so -1 is again a root. Now consider only the first three numbers. This means the reduced equation is $3x^2+5x+1=0$, which can be solved using quadratic formula $$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}=\frac{-5 \pm \sqrt{5^2-4(3)(1)}}{2(3)}=\frac{-5\pm\sqrt{13}}{6}$$So, we have obtained the two other roots.

anonymous · Answer

so what would the final answer be?

anonymous · Answer

and thank you sooooo so so much, you've been super helpful! xx

anonymous · Answer

Just collect the results. x = -1 (double root), x = $\frac{-5\pm\sqrt{13}}{6}$.

Your very much welcome. :)

anonymous · Answer

so it would be -1 multiplity 2, and then the fraction

anonymous · Answer

yep. And one thing, its "multiplicity".