zeros of a ploynomial function of the 5th degree?

Question

I am having trouble finding the 3 zeros (solutions) using synthetic division can someone please help me

4x^5 - 5x^4 - 91x^3 + 35x^2 + 237x - 60 = 0

Answer 1

There is no closed form solution to the zeros of a 5th order polynomial. All you can do is guess and check your guesses via synthetic division. You can make your guesses smarter by plotting a few points.

If you plot the function you will see there are 5 real zeros. Just by looking at the plot, the zeros are around -4, -1.75, 0.25, 1.75 and 5.

Try the integer guesses first (-4 and 5) just because they are the easiest to do. You will find that both are roots leaving:

4x^3 - x^2 - 12*x + 3 = 0

If you try the fractional roots, you'll find that x= 1/4 is a root but the other guesses are not. Dividing out x - 1/4 leaves:

x^2 - 3 = 0

So the final 2 roots are +sqrt(3) and -sqrt(3).


The complete factorization is:

(x + 4)(x - 5)(4x - 1)(x - sqrt(3))(x + sqrt(3))

Answer 2

I suggest you to use the rational root theorem
http://en.wikipedia.org/wiki/Rational_root_theorem

Assuming that you read the theorem from the link
provided, we consider the divisors of 4 and 60:

Divisors of 4: 1, 2, 4, -1, -2, -4
Divisors of 60: 1, 2, 3, 4, ..., 60, -1, -2, -3, -4, ..., -60
(there should be total 24 of them, but of course we don't
have to list all of them at the beginning since we can
just try some of small raional root candidate).

Let f(x) = 4x^5 - 5x^4 - 91x^3 + 35x^2 + 237x - 60.
I tried to compute f(1), f(-1), f(2), f(-2), f(3), and f(-3),
but none of them vanishes. Luckily we have f(-4) = 0.
Hence, x = -4 is a root to the polynomial. Thus, we
can write f(x) = (x - (-4))g(x) for some polynomial g(x)
which we can find by dividing f(x) by x + 4. After division,
we have g(x) = 4x^4 - 21x^3 - 7x^2 + 63x - 15.

Now we use the theorem again. Consider the divisors
of 4 and -15.

Divisors of 4: 1, 2, 4, -1, -2, -4
Divisors of 15: 1, 3, 5, 15, -1, -3, -5, -15

After few trials, we have g(5) = 0. Thus, g(x) = (x - 5)h(x)
where h(x) will be some polynomial again. We divide
g(x) by x - 5, then h(x) = 4x^3 - x^2 - 12x + 3.

Now h(x) is rather easy to factor:

h(x) = 4x^3 - x^2 - 12x + 3
= x^2(4x - 1) - 3(4x - 1)
= (4x - 1)(x^2 - 3)
= (4x - 1)(x - Sqrt[3])(x + Sqrt[3])

Thus, we get the roots x = 1/4, Sqrt[3], -Sqrt[3].

Therefore, the roots we found are
x = -4, 5, 1/4, Sqrt[3], and -Sqrt[3].

Answer 3

By graphing guessing and then testing in the equation. I find that there are zeros at 0.25, sqrt(3), -sqrt(3)


Of course since it is a 5th order equation there must be 5 zeros (real or imaginary). Without further work I cannot tell you which zero if any of the 3 repeats.

Answer 4

Since it is difficult to type synthetic division solution over here, I will have to ask you to look at the reference site for step by step instructions. (This site is very good)

I will just provide the solution here. You can refer to the site below for step by step instructions. If you need more help, reply.


(x - 5) (x + 4) (4 x^2 - 1) (x - 3)

Answer 5

with a scientific calculator I got
x=0.25
x=+-1.732055080757
x=-4
x=5