Respuesta :
A rational number is a number that can be written in the form p/q where q is not equal to zero and p is an integer.
A root is a value for x that makes the function equal to zero. It is also called a solution.
Hence, a rational root is such a number that makes any polynomial function equal to zero.
let us consider a general polynomial equation-
P(x) = anxn + an− 1xn − 1 + … + a1x1 + a0
hence, the rational roots must be of the form = [tex]\pm[/tex] (factors of a0) / (factors of an).
let us now take an example,
f(x)=[tex]2 x^3-3 x^2-11 x+6[/tex]
here ao = 6, and factors of 6: 1,2,3,6
an = 2 and the factors of 2: 1,2
Hence, we have Possible Rational Roots:
[tex]$\pm\left\{\frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{2}{2}, \frac{3}{1}, \frac{3}{2}, \frac{6}{1}, \frac{6}{2}\right\}$[/tex]
removing the duplicate values we have- [tex]$\pm\left\{1, \frac{1}{2}, 2, 3, \frac{3}{2}, 6\right\}$[/tex]
In this way, we have the rational numbers which can be possible solutions for our function f(x).
Now we substitute each of these rational numbers one by one to check if it satisfies the equation as follows.
f(1) =[tex]2 (1)^3-3 (1)^2-11 (1)+6[/tex] = 2-3-11+6 = -6 [tex]\neq[/tex] 0.
Hence, it doesn't satisfy the equation, it means it's not the root of the equation.
Similarly, we can check for the rest of the numbers.
we will get f(1/2)=0, f(3)=0 and f(-2)=0
Hence, x=3,1/2 or -2 are the solutions for the equation f(x).
Read more about Rational roots:
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