We can find a few roots of some polynomial equations by trial and error method.

**Polynomial Equations with no Additional Information**

**Rational Root Theorem**

We can find a few roots of some polynomial equations by trial and
error method. For instance, we consider the equation

4*x*^{3} âˆ’ 8*x*^{2} âˆ’ *x *+ 2 = 0 ... (1)

This is a third degree equation which cannot be solved by any
method so far we discussed in this chapter. If we denote the polynomial in (1)
as *P*(*x*) , then we see that *P*(2) = 0 which says that *x
*âˆ’ 2 is a factor. As the
rest of the problem of solving the equation is easy, we leave it as an
exercise.

**Example 3.25**

Solve the equation *x*^{3} âˆ’ 5*x*^{2} âˆ’ 4*x *+ 20 = 0
.

**Solution**

If *P*(*x*) denotes the polynomial in the
equation, then *P*(2) = 0 . Hence 2 is a root of the polynomial equations.
To find other roots, we divide the given polynomial *x*^{3} - 5*x*^{2} - 4*x *+
20 by *x *- 2 and get *Q*(*x*)= *x*^{2} - 3*x *-10 as
the quotient. Solving *Q*(*x*) = 0 we get -2 and 5 as roots. Thus 2,
-2, 5 are the solutions of the given equation.

Guessing a number as a root by trial and error method is not an
easy task. But when the coefficients are integers, using its leading
coefficient and the constant term, we can list certain rational numbers as
possible roots. Rational Root Theorem helps us to create such a list of
possible rational roots. We recall that if a polynomial has rational
coefficients, then by multiplying by suitable numbers we can obtain a
polynomial with integer coefficients having the same roots. So we can use **Rational Root Theorem**, given below, to
guess a few roots of polynomial with rational coefficient. We state the theorem without
proof.

**Theorem 3.5 (Rational Root Theorem)**

Let *a*_{n} *x*^{n} +...+ a_{1}*x* + a_{0} with a_{n} â‰ 0, and a_{0} â‰ 0, be a polynomial with integer coefficients. If *p*/*q*,
with ( *p, q*) = 1, is a root of the polynomial, then *p* is a factor
of a_{0} and *q* is a
factor of a_{n} .

When a_{n} = 1, if there is a rational root p/q, then as per theorem 3.5 q
is a factor of a_{n} , then we must have q = Â±1.Thus *p* must be an integer. So a
monic polynomial with integer coefficient cannot have non-integral rational
roots. So when a_{n} = 1, if at all there is a rational root, it must be an integer
and the integer should divide a_{0}. (We say an integer *a* divides an integer
*b* , if *b* = *ad* for some integer *d*.)

As an example let us consider the equation *x*^{2} - 5*x *- 6 = 0 .
The divisors of 6 are Â± 1, Â± 2, Â± 3, Â± 6. From Rational Root Theorem, we can conclude
that Â±1, Â± 2, Â± 3, Â± 6 are the only possible solutions of the equation. It does
not mean that all of them are solutions. The two values -1 and 6 satisfy the
equation and other values do not satisfy the equation.

Moreover, if we consider the equation *x*^{2} + 4 = 0 ,
according to the Rational Root theorem, the possible solutions are Â±1, Â± 2, Â± 4; but none
of them is a solution. The Rational Root Theorem helps us only to guess a solution and
it does not give a solution.

**Example 3.26**

Find the roots of 2*x*^{3} + 3*x*^{2} + 2*x *+ 3 = 0 .

**Solution**

According to our notations, a_{n} = 2 and a_{0 }= 3 . If p/q is a zero of the polynomial, then
as ( p, q) = 1, p must divide 3
and q must divide 2. Clearly, the possible values of p are 1,
-1, 3, -3 and the possible values of q are 1, -1, 2, -2 . Using these
p and q
we can form only the fractions Â±1/1 ,
Â± 1/2 , Â± 3/2 , Â± 3/1.
Among these eight possibilities, after verifying by substitution, we get
-3/2 is the only
rational zero. To find other zeros, we divide the given polynomial 2*x*^{3} + 3*x*^{2} + 2*x *+ 3 by 2*x
*+ 3 and get *x*^{2} +1 as the quotient with zero remainder. Solving *x*^{2} +1 = 0 ,we get *i *and
-*i *as roots. Thus -3/2 - *i*, *i *are the roots of the given
polynomial equation.

Tags : Polynomial Equations with no Additional Information , 12th Mathematics : UNIT 3 : Theory of Equations

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12th Mathematics : UNIT 3 : Theory of Equations : Rational Root Theorem | Polynomial Equations with no Additional Information

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