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

Chapter: 12th Mathematics : UNIT 3 : Theory of Equations

Rational Root Theorem

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

4x3 8x2 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 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 x3 − 5x2 − 4x + 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  x3  - 5x2 - 4x + 20  by x - 2 and get Q(x)=  x2 - 3x -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 an xn +...+ a1x + a0 with an ≠ 0, and a0 ≠ 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 a0 and q is a factor of an .

When an = 1, if there is a rational root p/q, then as per theorem 3.5 q is a factor of an , then we must have q = ±1.Thus p must be an integer. So a monic polynomial with integer coefficient cannot have non-integral rational roots. So when an = 1, if at all there is a rational root, it must be an integer and the integer should divide a0. (We say an integer a divides an integer b , if b = ad for some integer d.)

As an example let us consider the equation x2 - 5x - 6 = 0 . The divisors of 6 are ± 1,  ± 2, ± 3, ± 6.  From Rational Root Theorem, we can conclude that ±1, ± 2, ± 3, ± 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  x2  + 4 = 0 , according to the Rational Root theorem, the possible solutions are ±1, ± 2, ± 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 2x3 + 3x2 + 2x + 3 = 0 .

Solution

According to our notations, an = 2 and a0 = 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 ±1/1 ,  ± 1/2 ,  ± 3/2 ,  ± 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 2x3 + 3x2 + 2x + 3 by 2x + 3 and get x2 +1 as the quotient with zero remainder. Solving x2 +1 = 0 ,we get i and -i as roots. Thus -3/2 - i, i are the roots of the given polynomial equation.

 

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

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12th Mathematics : UNIT 3 : Theory of Equations


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