Rational Root Theorem
The rational root theorem, as its name suggests, is used to find the rational solutions of a polynomial equation (or zeros or roots of a polynomial function). The solutions derived at the end of any polynomial equation are known as roots or zeros of polynomials. A polynomial doesn't need to have rational zeros. But if it has rational roots, then they can be found by using the rational root theorem.
Let us see the rational root theorem statement along with its proof. Also, we will solve some problems using the same.
What is the Rational Root Theorem?
The rational root theorem is also known as the rational zero theorem (or) the rational zero test (or) rational test theorem and is used to determine the rational roots of a polynomial function. The general form of a polynomial function is f(x) = \(a_{n}x^{n}+a_{n1}x^{n1}+.......+a_{2}x^2+a_{1}x+a_{0}\). Here, the value(s) of x that satisfy the equation f(x) = 0 are known as the roots (or) zeros of the polynomial. For example, x = 1, 2, and 3 are the zeros of the cubic function f(x) = x^{3}  4x^{2}^{ }+ x + 6 (one can easily check this by substituting each zero in the polynomial and see f(x) = 0). But how to find these zeros? Let's see.
Rational Zero Theorem Statement
The rational zero theorem states that each rational zero(s) of a polynomial with integer coefficients f(x) = \(a_{n}x^{n}+a_{n1}x^{n1}+.......+a_{2}x^2+a_{1}x+a_{0}\) is of the form p/q, where
 p is a factor of the constant \(a_0\) and
 q is a factor of the leading coefficient \(a_n\).
 p and q are relatively prime (i.e., the fraction p/q is in its simplest form)
Rational Zero Theorem Proof
To prove the rational root theorem, we will assume that p/q is a rational zero of the polynomial f(x) = \(a_{n}x^{n}+a_{n1}x^{n1}+.......+a_{2}x^2+a_{1}x+a_{0}\) and we will prove that p is a factor of \(a_0\) and q is a factor of \(a_n\). So assume that p/q is a rational number which is a zero of a polynomial f(x), where p and q are relatively prime numbers and q ≠ 0.. Then x = p/q satisfies the equation f(x) = 0. i.e.,
\(a_{n}\left(\dfrac{p}{q}\right)^{n}+a_{n1}\left(\dfrac{p}{q}\right)^{n1}+.......+a_{2}\left(\dfrac{p}{q}\right)^2+a_{1}\left(\dfrac{p}{q}\right)+a_{0}\) = 0
Multiplying both sides by q^{n},
\(a_n\) p^{n} + \(a_{n1}\) p^{n1} q + ... + \(a_2\) p^{2} q^{n2} + \(a_1\) p q^{n1} + \(a_0\) q^{n} = 0 ... (1)
Proving that p is a factor of a₀
Subtracting \(a_0\) q^{n} from both sides of (1),
\(a_n\) p^{n} + \(a_{n1}\) p^{n1} q + ... + \(a_2\) p^{2} q^{n2} + \(a_1\) p q^{n1} = \(a_0\) q^{n}
One can easily see that p divides each term on the left side and hence p divides the entire sum that is on the left side. Hence p divides the right side also (as left side and right side are connected by "equal to" sign). i.e.,
p is a factor of \(a_0\) q^{n}.
Since p and q are relatively prime numbers, p divides \(a_0\) (or) p is a factor of \(a_0\).
Proving that q is a factor of aₙ
Subtracting \(a_n\) p^{n} from both sides of (1),
\(a_{n1}\) p^{n1} q + ... + \(a_2\) p^{2} q^{n2} + \(a_1\) p q^{n1} + \(a_0\) q^{n} = \(a_n\) p^{n}
One can easily see that q divides each term on the left side and hence q divides the entire sum that is on the left side. Hence q divides the right side also (as left side and right side are connected by "equal to" sign). i.e.,
q is a factor of \(a_n\) p^{n}.
Since p and q are relatively prime numbers, q divides \(a_n\) (or) q is a factor of \(a_n\).
Hence the rational root theorem is proved.
Listing Possible Rational Zeros Using Rational Root Theorem
The rational zero theorem is used to find the list of all possible rational zeros of a polynomial f(x). Here, the word "possible" means that all the rational zeros provided by the rational root theorem need NOT be the actual zeros of the polynomial. Here are the steps to find the list of possible rational zeros (or) roots of a polynomial function. The steps are explained through an example where we are going to find the list of all possible zeros of a polynomial function f(x) = 2x^{4}  5x^{3 } 4x^{2} + 15 x  6.
 Step  1: Identify the constant and find its factors (both positive and negative). These factors would give the possible values of p.
Here, the constant is 6 and its factors are, p = ± 1, ± 2, ± 3, and ± 6.  Step  2: Identify the leading coefficient and find its factors (both positive and negative). These factors would give the possible values of q.
Here, the constant is 2 and its factors are, q = ± 1 and ± 2.  Step  3: Find each possible value of p/q (find all combinations by dividing every value of p by every value of q) in the simplest form.
When q = ± 1, p/q = ±1/±1, ± 2/±1, ± 3/±1, ± 6/±1 = ± 1, ± 2, ± 3, ± 6.
When q = ± 2, p/q = ±1/±2, ± 2/±2, ± 3/±2, ± 6/±2 = ± 1/2, ± 1, ±3/2 , ± 3.  Step  4: List all the possible rational zeros from Step  3 by removing the duplicates.
Then the list of all possible rational zeros of f(x) = 2x^{4}  5x^{3 } 4x^{2} + 15 x  6 are ± 1, ± 2, ± 3, ± 6, ± 1/2, and ± 3/2.
Finding All Zeros Using Rational Root Theorem
In the previous section, we have seen how to find the list of possible zeros of a polynomial function. But all the numbers from the list may not be the actual zeros. We can find the actual rational zeros by using the remainder theorem (i.e., by substituting each zero in the given polynomial and see whether f(x) = 0). Once we find the rational zeros, sometimes it is possible to find the other roots (irrational roots or complex roots) as well. Here are the steps for the same. In these steps, we will find all the zeros of the same polynomial (as in the previous section) f(x) = 2x^{4}  5x^{3 } 4x^{2} + 15 x  6.
 Step  1: Find the list of all possible rational zeros using the rational zero theorem.
From the previous section, the list of all possible rational zeros of f(x) = 2x^{4}  5x^{3 } 4x^{2} + 15 x  6 are ± 1, ± 2, ± 3, ± 6, ± 1/2, and ± 3/2.  Step  2: Substitute each of the numbers from Step  1 in f(x) and check which number would result in f(x) = 0. This will give the actual rational zeros.
After checking with each number, we can easily see that f(1/2) = 0 and f(2) = 0. This is because
f(1/2) = 2(1/2)^{4}  5(1/2)^{3 } 4(1/2)^{2} + 15 (1/2)  6 = 0
f(2) = 2(2)^{4}  5(2)^{3 } 4(2)^{2} + 15(2)  6 = 0  Step  3: Divide the given polynomial by each of the actual rational zeros from Step  2. The synthetic division makes this process easier.
 Step  4: Find the quotient of the above division (using the last line) and set it to zero. Solve the resultant equation to find the other zeros.
2x^{2} + 0x  6 = 0
2x^{2} = 6
x^{2} = 3
x = ±√3
Thus, all the roots (or zeros) of f(x) = 2x^{4}  5x^{3 } 4x^{2} + 15 x  6 are 1/2, 2, √3 and √3.
Applications of Rational Root Theorem
Here are the applications/uses of the rational root theorem.
 The rational root theorem gives all the possible rational zeros of the polynomial.
 The possible zeros can be verified to check whether they are the actual roots by substituting into the polynomial.
 Finding the rational zeros may help in finding the irrational zeros/complex zeros after using synthetic division.
 Finding the possible zeros may help to identify the graph of a polynomial (among a group of graphs).
Important Notes on Rational Root Theorem:
 The possible rational zeros of a polynomial function are given by all possible p/q values where p is a factor of the constant and q is a factor of leading coefficient.
 While finding the values of p and q, we have to consider both poisitive ane negative factors.
 If the leading coefficient of a polynomial is 1, then the factors of the constant themseveles are the possible rational zeros of f(x).
 The rational zero theorem helps us to find the zeros of a polynomial function only if it has rational zeros.
 The rational zero theorem helps in solving polynomial equations.
Related Topics:
Rational Root Theorem Examples

Example 1: Find the possible rational zeros of the cubic function f(x) = 3x³  5x² + 4x + 2.
Solution:
The constant term is 2 and its factors are ± 1 and ± 2. These would be the values of p.
The leading coefficient is 3 and its factors are ± 1 and ± 3. These would be the values of q.
Then by the rational zero theorem, the possible rational roots of f(x) are all possible values of p/q.Values of p/q when q = ± 1:
p/q = ± 1/ ±1, ±2 / ±1 = ± 1, ± 2.Values of p/q when q = ± 3:
p/q = ± 1/ ±3, ±2 / ±3 = ± 1/3, ± 2/3Answer: The possible rational zeros of f(x) are ± 1, ± 2, ± 1/3, and ± 2/3.

Example 2: Find the actual rational zeros of the cubic function that is given in Example 1.
Solution:
We have already found the possible rational zeros of f(x) by using rational zero theorem in Example 1 to be ± 1, ± 2, ± 1/3, and ± 2/3.
We just substitute each of these in the equation f(x) = 0 and see which will satisfy. f(1) = 3(1)³  5(1)² + 4(1) + 2 = 10
 f(1) = 3(1)³  5(1)² + 4(1) + 2 = 4
 f(2) = 3(2)³  5(2)² + 4(2) + 2 = 50
 f(2) = 3(2)³  5(2)² + 4(2) + 2 = 14
 f(1/3) = 3(1/3)³  5(1/3)² + 4(1/3) + 2 = 0
 f(1/3) = 3(1/3)³  5(1/3)² + 4(1/3) + 2 = 26/9
 f(2/3) = 3(2/3)³  5(2/3)² + 4(2/3) + 2 = 34/9
 f(2/3) = 3(2/3)³  5(2/3)² + 4(2/3) + 2 = 10/3
Since we have got f(1/3) = 0, x = 1/3 is a rational zero.
Answer: The only rational zero of the given cubic function is 1/3.

Example 3: Find all the zeros of the cubic function that is given in Example 1.
Solution:
From Example 2, we found that the rational zero of f(x) is 1/3.
Let us divide the given polynomial by x = 1/3 (or we can say that we have to divide by 3x + 1) using synthetic division.
Now, set the quotient equal to 0 to find the other zeros.
3x²  6x + 6 = 0
Divide both sides by 3,
x²  2x + 2 = 0
Using quadratic formula,
x = \(\dfrac{b \pm \sqrt{b^{2}4 a c}}{2 a}\)
x = \(\dfrac{(2) \pm \sqrt{(2)^{2}4 (1) (2)}}{2 (1)}\)
= \(\dfrac{2 \pm \sqrt{4}}{2 }\)
= \(\dfrac{2 \pm 2i}{2 }\)
= 1 ± iAnswer: The zeros of f(x) are 1/3, 1 + i and 1  i.
FAQs on Rational Root Theorem
What is Rational Root Theorem Definition?
The rational root theorem says, a rational zero of a polynomial is of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
What is the Other Name of Rational Zero Test?
The rational zero test is also known as the "rational zero theorem" (or) "rational root theorem". It is used to find the list of possible rational zeros of a polynomial function.
Why is the Rational Root Theorem Useful?
Rational root theorem is used to find the set of all possible rational zeros of a polynomial function (or) It is used to find the rational roots (solutions) of a polynomial equation.
How Do You Find Possible Rational Zeros Using Rational Zero Theorem?
To find the rational zeros of a polynomial function f(x),
 Find the constant and identify its factors. Each number represents p.
 Find the leading coefficient and identify its factors. Each number represents q.
 Find all possible combinations of p/q and all these are the possible rational zeros.
 All these may not be the actual roots. To find the actual rational zeros, just substitute see which of them satisfies f(x) = 0.
What is the Rational Root Theorem Equation?
According to the rational root theorem, the rational zero of a polynomial f(x) is of the form p/q where p and q are factors of the constant and leading coefficient respectively.
What Do p and q Stand for in the Rational Zero Theorem?
In the rational zero theorem, p and q stand for all potential rational roots of a polynomial. p represents all positive and negative factors of the constant of the polynomial whereas q represents all positive and negative factors of the leading coefficient of the polynomial.
What is the Process of Finding Rational Zeros?
For finding the rational zeros of a polynomial function, just find all possible values of p/q where p is a factor of the constant of polynomial and q is a factor of the leading coefficient of the polynomial. Then we get a set of numbers. Substitute each number in the polynomial (or divide the polynomial by each number by synthetic division) and see which one would result in 0.
Are Rational Roots and Rational Zeros the Same?
Rational roots are also known as rational zeros. For a polynomial, it is considered the same as finding the rational xintercepts.
How Do You Use the Rational Zero Theorem?
The rational zero theorems can also be called the rational zero theorem. Here are the simple steps that can be used:
 Arrange f(x) in descending order of exponents of the variable.
 Write the factors of the constant term i.e. all the possible values of p.
 Write the factors of the leading coefficient i.e. all the possible values of q.
 Write the possible values of p/q. Remember that since factors can be negative, p/q and  p/q must both be included. Simplify each value and cross out any duplicates.
 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).
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