How do you know how many zeros are in a polynomial function?

Question

Do you go by the leading coefficient or the multiplicity..

Ex...3x^4+7x^3+x^2-7x-4

How many zeros...I got three (1, -1 and -4/3)...so does that mean we go by leading coef...basaically, how can u tell?

THANKS!

Answer 1

since the degree of your equation is 4, you will have 4 solutions

The sum of your solutions = - 7/3

1 - 1 - 4/3 + x = - 7/3 so your last root is -1. In other words, -1 is a double root of your equation.

Answer 2

In your case since the polynomial is of 4th degree there are theoretically 4 zeros of which you found 3 distinct zeros. Divide your polynomial by (x-1)(x+1)(x+4/3) and find the zero if the result to give you the 4th. It may or may not be distinct. (It may be the same as one of the three you already have)

Answer 3

The number of roots (including all real and complex roots) of a polynomial equation in one variable is generally equal to the highest power of the variable in the function.

However, there are exceptions: if the function happens to be a perfect square, two of the roots will be equal; if it is a perfect cube, three will be equal, and so forth. eg. x**2 - 2*x + 1 = 0. If we try to factorise, we see this is a perfect square; (x - 1) * (x - 1) = 0 so x = 1 and there isn't another solution.

Answer 4

In general a polynomial of grade n has n zeros wihch can be real or complex. If one complex solution (zero) exists it exists also its complex conjugate i.e.
if a + jc is a solution also a -jc is a solution
A plolynomial can also bed written in the format
(x-x0)*(x-x1)*..*(x-xn)
where x0..xn are the solution
There is no resolving formula for polynomial above 4th grade. However if you happen to know 1 solution you can get a polnymomial of grade n-1 by dividng the original polynomial by (x-x0) where x0 is the knwon zero.

Answer 5

check the highest degree, in your case , the highest degree is 4 , therefore it has 4 roots.
(x+1)(-3x-4)(x-1)^2= function.the roots are -1,1,4/3 but double root on 1

Answer 6

The number of zeroes is equal to the order of the polynomial; in your example it is four.