Rational Zero Theorem

                                 Rational Zero Theorem 

If P(x) is a polynomial with integer coefficients such that p is a factor of the constant term and q is a factor of the leading coefficient then $\frac{p}{q}$ is a zero of P(x) and $P(\frac{p}{q})=0$. 

It is the Rational Zero Theorem. It helps us to find all the possible zeros of a polynomial. Even we can examine and state whether a given number is a zero of polynomial or not. 

Let's take an example. Consider $P(x)=2x^3-5x^2-4x+3$ 

Here the constant term is $3$  and all the factors of $3$ are $-1, 1, 3, -3$. 

Therefore, p can take any value from the set $\text{-3, -1, 1, 3}$.
And leading coefficient is 2 and all the factors of 2 are $-1, -2, 1, 2$
Therefore, q can take any value from the set ${-2, -1, 1, 2}$.
Now all the possible zeros of P(x) are listed as $\pm \frac{p}{q}$ 
We can list them as $\pm\frac{1}{1},\pm\frac{1}{2},\pm\frac{3}{1},\pm\frac{3}{2}$ 

Or we can simplify them as $\pm1,\pm3,\pm\frac{1}{2},\pm\frac{3}{2}$
These are all the possible zeros of the given P(x) guaranteed by Rational Zero Theorem. 
We can now use Remainder Theorem to identify which one is actually a zero of the given polynomial P(x). 

According to Remainder Theorem, If P(a)=0 where a be a number then a is known as zero of P(x). Let's take x=1, a zero  guaranteed by the Rational Zero Theorem,
We have $P(1)=2\times 1^3-5\times1^2-4\times 1+3=2-5-4+3=-4$ 

Since $P(1)\ne 0$, hence $x=1$ is not a zero of P(x). 

Let's take $x=-1$, we have $P(-1)=2(-1)^3-5(-1)^2-4(-1)+3=-2-5+4+3=0$. 
Since $P(-1)= 0$, hence $x=-1$ is a zero of P(x) too. 

Now let's take next possible zero,guaranteed by Rational Zero Theorem,  x=3 we will plug in 3 for x in P(x) and check whether it gives us zero or not. 

$P(3)=2\times 3^3-5\times 3^2-4\times 3+3=54-45-12+3=0$

We see that $P(3)=0$, hence x=3 is a zero of P(x) too. 

Similarly, we can check whether a number is a zero of the given polynomial or not.
Quick check: A number is a zero of the given polynomial if it's included in the rational zeros guaranteed by Rational Zero Theorem.
Secondly, we can use Remainder Theorem or Synthetic Division Method  to check whether a number is a zero of polynomial or not.
Try these problems: 
1. Find all the possible rational zeros of the polynomial $P(x)=5x^3-3x^2-2x+9$ guaranteed by Rational Zero Theorem.
2. Solve the equation $6x^3+31x^2+3x-10=0$