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:

It subtracts sadness and adds happiness in our life.

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”





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