## Rational zeros theorem

Rational zeros theorem gives the possible rational zeros of a polynomial function. Equivalently the theorem gives all the possible roots of an equation.

The theorem states that,

If f(x) = anxn+an-1xn-1+…. +a1x+a0 has integer coefficients and p/q(where p/q is reduced) is a rational zero, then .p is the factor of the constant term a0 and q is the factor of leading coefficient an.

The steps to be followed:

1) We have to arrange the polynomial in descending order.

2) Write down all the factors of the constant term. Those are all the possible values of p.

3) Write down all the factors of the leading coefficient which are all possible values of q.

4) We have to write down all the possible values of p/q.

5) We have to use synthetic division to determine the values of p/q for which f(p/q) =0. These are all the rational roots of f(x).

Example: List all possible rational zeros of f(x) = x3-2x2-x+2 using rational zeros theorem.

Solution:

1) The equation is already arranged in descending order.

2) Factors of the constant term 2 is = ±1, ±2

3) Factors of the leading coefficient is = ±1.

4) Possible values of p/q = ±1/±1, ±2/±1.

These can be simplified into ±1, ±2.

5) Now we have to use synthetic division to check all the possible zeros.

We have another shortcut method to check whether 1 and -1 are the factors.To check that we have to follow these instructions.
(i) If the sum of all coefficients in a polynomial including the constant term is zero, then x-1 is a factor.
(ii) If the sum of the coefficients of even powers together with the constant term is the same as the sum of the coefficients of odd powers, then (x+1) is a factor. Rational zeros theorem

### Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life. They are: