Rational Root Theorem also called Rational Zero Theorem in algebra is a systematic approach of identifying rational solutions to polynomial equations.
According to Rational Root Theorem, for a rational number to be a root of the polynomial, the denominator of the fraction must be a factor of the leading coefficient (the coefficient of the term with the highest power of the variable). Additionally, the numerator of the fraction must be a factor of the constant term (the term that doesn’t include the variable). This theorem is useful for narrowing down possible rational solutions to a polynomial equation.
Rational Root Theorem helps in the quick identification of rational solutions of polynomial equations. We can also find roots by using a specific formula or by factorizing the polynomial. In this article, we will discuss about rational root theorem in detail, with its examples, formula, and some solved examples to understand the concept of the Rational Root Theorem.
What is the Rational Root Theorem?
The Rational Root Theorem in algebra helps us find possible rational solutions to polynomial equations with whole number coefficients. Roots of a Polynomial are those specific values for which the polynomial yields zero as the value of the polynomial. To have a rational solution in such an equation, we need to satisfy two conditions:
- The leading coefficient (the one with the highest power of x) must be divisible by the denominator of the fraction.
- The constant term (the part without any x) must be divisible by the numerator.
The Rational Root Theorem provides a way to determine all possible rational roots of a polynomial equation. The formula states that any rational solution, expressed in its simplest form as 𝑝𝑞qp, to the polynomial equation
a0 + a1x + a2x2 + … + anxn
Where a0, a1, …, an are just regular whole numbers, to find a rational solution p/q, q must divide an, and p must divide a0.
- p (the numerator of the fraction) must be a factor of the constant term a0.
- q (the denominator of the fraction) must be a factor of the leading coefficient an.
Rational Root Theorem Definition
Rational Root Theorem is a method of identifying rational solutions to polynomial equations. According to Rational Root Theorem, the possible rational roots of a polynomial is given by the combination of ratio of all the possible divisors of the constant terms and the leading coefficient. Roots of a polynomial can be found by equating the polynomial with zero. The roots can be rational or irrational. To have a rational solution in such an equation, we need to satisfy two conditions as mentioned above:
- The leading coefficient must be divisible by the denominator of the fraction.
- The constant term must be divisible by the numerator.
Given polynomial equation like this:
a0 + a1x + a2x2 + … + anxn
Here, the rational solution will be p/q given q must divide a0 and p must divide an i.e. p is the factor of a constant term and q is the factor of leading coefficient.
How to Find Rational Zeros?
Rational Root Theorem establishes two key conditions that must be met for a polynomial equation to have rational solutions:
- Leading Coefficient Condition: The leading coefficient, which is the coefficient of the term with the highest power of the variable, must be divisible by the denominator of the fraction representing the rational solution.
- Constant Term Condition: The constant term, which is the part of the equation without any variable, must be divisible by the numerator of the fraction representing the rational solution.
Understanding the rational solutions is essential in algebra and can help us solve various mathematical problems. While basic, this concept provides a strong foundation for more advanced math and problem-solving techniques.
Also Read: Zeros of Polynomial
Rational Root Theorem Example
Let’s consider the example equation:
2x2 – 5x + 1
- Here, we need to find rational solutions, which are fractions in the form p/q. The leading coefficient is 2, and the constant term is 1.
- For this equation to have rational solutions, q must divide 2, and p must divide 1.
- As 1, has no divisors other than 1, and 2 can be divided by 1 or 2, the possible rational solutions are limited to 1/1 or 1/2 (0.5).
So, the equation can have solutions like x = 1 or x = 1/2
Note here that we aim to discover rational solutions in the form of a p/q
Rational Root Theorem Proof
The Rational Root Theorem provides a technique for identifying potential rational solutions of a polynomial equation.
It states that if a rational solution, m/n, exists for the equation, then m must be a divisor of the constant term, and n must be a divisor of the leading coefficient.
This theorem is a valuable instrument for efficiently narrowing down the hunt for rational solutions to polynomial equations.
Here, we will consider p/q as a rational zero of a given polynomial f(x).
a0 + a1x + a2x2 + … + anxn
or
anxn + an-1xn-1 +…… + a2x2 + a1x + a0
rational solution will be p/q given that q must divide an and p must divide a0.
Considering p/q is a zero and rational solution of which p and q have no common factor and q ≠ 0.
Now, since p/q is a zero of polynomial equation.
an x n + an-1 x n-1 +…… + a2 x 2 + a1 x + a0 = 0
Put (p/q) as a root solution
an(p/q)n + an-1(p/q)n-1 +…… + a2(p/q)2 + a1(p/q) + a0 = 0
Multiplying both sides by qn
an(p)n + an-1(p)n-1(q)n +…… + a2(p)2(q)n-1 + a1(p)(q)n-1 + a0qn = 0 (Consider it as eq. 1)
Proving p is a factor of a0
To prove p is a factor of a0, we will subtract a0qn from both sides of eq. 1
an(p)n + an-1(p)n-1(q)n +…… + a2(p)2(q)n-1 + a1(p)(q)n-1 = – a0qn (Consider it as eq. 2)
Since p is a factor of every term on the left side given left side is equal to right side p will also be a factor of right side and given p, q have no factor in common so p will be factor of a0, hence proved.
Proving q is a factor of an
To prove q is a factor of an, we will subtract anpn from both sides of eq. 1
an-1(p)n-1(q)n +…… + a2(p)2(q)n-1 + a1(p)(q)n-1 + a0qn = – an(p)n
Since q is a factor of every term on the left side given left side is equal to right side q will also be a factor of the right side and given p, q have no factor in common so p will be factor of an, hence proved.
How to Find Zeros using Rational Zero Theorem?
The Rational Root Theorem is a valuable tool in algebra to help find the possible rational roots (or zeros) of a polynomial equation. It can significantly simplify the process of solving for the roots of a polynomial equation.
Steps to Find Rational Zero
- List Possible Numerators: Find all the factors of the constant term (a0) in the polynomial equation. These are the potential numerators of rational roots.
- List Possible Denominators: Find all the factors of the leading coefficient (an) in the polynomial equation. These are the potential denominators of rational roots.
- Test the Combinations: Combine each numerator with each denominator to create pairs. These pairs represent potential rational roots. For each pair, substitute the value into the polynomial equation and check if it equals zero.
- Identify the Real Roots: The roots that make the polynomial equation equal to zero are the real zeros of the polynomial.
Below is an example for illustration of rational theorem to Find Zeros using Rational Zero Theorem :
In our illustration, we have the polynomial 3x3 − 2x2 − 8x + 4
Rational theorem suggests that any rational solution will take the form m/n, where p is a divisor of the constant term (in this instance, 4), and q is a divisor of the leading coefficient (here 3). Hence, the values for m can be ±1, ±2, ±4, all of which are divisors of 4, and the values for n can be ±1, ±3 which are divisors of 3.
The possible combination of roots will be ±1/ ±1, ±2/±1, ±4/±1, ±1/ ±3, ±2/±3, ±4/±3
Now, we will be using the zero values and generate various fractions p/q as potential solutions for polynomial equation. Let’s test these possible solutions in the equation:
When x = 1/1, we calculate = 3(1)3 −2(1)2 − 8(1) + 4 = 3 − 2 − 8 + 4 = −3
When x = −1/1, we obtain 3(−1)3 − 2(−1)2 − 8(−1) + 4 = −3 − 2 + 8 + 4 = 7
Regrettably, none of these values makes the equation equal to zero. Now we can check other combinations of m/n if they are root or not.
By applying the Rational Root Theorem, you can quickly identify potential rational roots and determine the real zeros of a polynomial equation, simplifying the process of solving for these critical values.
Applications of Rational Root Theorem
The applications of Rational Zero Theorem are listed below:
- The Rational Root Theorem is a powerful tool for finding potential rational solutions to polynomial equations.
- To apply the theorem, you first identify the leading coefficient and the constant term of the polynomial equation. Then, you list the divisors of both coefficients.
- Potential rational solutions are fractions formed by pairing the divisors in the form p/q, where p is a divisor of the constant term, and q is a divisor of the leading coefficient.
- Rational Root Theorem aids mathematicians in solving various mathematical problems, including those in the physical sciences.
- Rational Root Theorem forms the foundation for more advanced calculus and algebraic techniques.
- While more advanced mathematical techniques and computing methods have emerged, the Rational Root Theorem remains a valuable concept taught in introductory mathematics courses. It provides a foundational understanding of rational solutions to polynomial equations, which is essential for building more complex mathematical skills.
People Also Read:
Solved Examples on Rational Root Theorem
Example 1: What are the possible rational roots of 2x3 – 7x2 – 2x + 4?
Solution:
Leading Coefficient Condition:
The leading coefficient is 2, and its divisors are ±1 and ±2.
Constant Term Condition:
The constant term is 4, and its divisors are ±1 ±2 and ±4.
Based on the Rational Root Theorem, potential rational solutions include ±1, ±2, or ±4 as numerators, and ±1 or ±2 as denominators.
Hence, possible rational zeros of given polynomial is ±1/±1, ±2/±1, ±4/±1, ±1/±2, ±2/±2, ±4/±2. Now after removing the duplicate combinations we will have 1, -1, 2, -2, 1/2, -1/2, 4, -4
Example 2: What are the possible rational zeros of the : x3 – 4x2 + 4x – 1?
Solution:
Leading Coefficient Condition:
The leading coefficient is 1, and its only divisor is 1.
Constant Term Condition:
The constant term is -1, and its divisors are 1 and -1
According to the Rational Root Theorem, potential rational solutions include ±1 as numerators and ±1 as denominators. Hence, possible zeros are 1 and -1
Let’s test these potential solutions in the equation:
When x = 1/1, we get: (1)3 – 4(1)2 + 4(1) – 1 = 1 – 4 + 4 – 1 = 0.
x = 1 is a rational solution that satisfies the equation.
Now for x = -1
(-1)3 – 4(-1)2 + 4(-1) – (-1) = -1 – 4 – 4 + 1 = -8
Hence, -1 is not the zero of the given polynomial.
Thus only 1 is the polynomial of given equation
Example 3: Find all the possible rational zeros for the polynomial 2x4 − 5x3 − 3x2 + 6x + 4
Solution:
Leading Coefficient Condition: The leading coefficient is 2, and its divisors are ±1 and ±2.
Constant Term Condition: The constant term is 4, and its divisors are ±1 ±2 and ±4.
According to the Rational Root Theorem, potential rational solutions include numerators of ±1, ±2, ±4 and denominators of ±1, ±2.
Based on the Rational Root Theorem, potential rational solutions include ±1, ±2, or ±4 as numerators, and ±1 and ±2 as denominators.
Hence, possible rational zeros of given polynomial is ±1/±1, ±2/±1, ±4/±1, ±1/±2, ±2/±2, ±4/±2. Now after removing the duplicate combinations we will have 1, -1, 2, -2, 1/2, -1/2, 4, -4
Example 4: Find the possible rational zeros of the given polynomial 4x5 + 2x4 − 6x3 + 3x2+ 1 = 0
Solution:
Leading Coefficient Condition: The leading coefficient is 4, and its divisors are ±1, ±2 and ±4.
Constant Term Condition: The constant term is 1, and its divisor is ±1.
According to the Rational Root Theorem, potential rational solutions include numerators of ±1 and denominators of ±1, ±2 and ±4.
Hence, possible rational zeros of the equation are ±1 /±1 , ±1/±2, ±1/±4
Example 5: What are the rational solutions of polynomial 3x2 − 2x − 5
Solution:
Leading Coefficient Condition: The leading coefficient is 3, and its divisors are ±1 and ±3.
Constant Term Condition: The constant term is -5, and its divisors are ±1 and ±5.
Based on the Rational Root Theorem, potential rational solutions comprise numerators of ±1, ±5 and denominators of ±1, ±3.
Hence As per Rational Zeros Theorem, the possible rational zeros of the given polynomial are ±1/±1, ±1/±3, ±5/±1, ±5/±3
Rational Root Theorem Worksheet
Q1: Find all the rational solutions of the equation: 3x3 – 10x2 – 11x + 4 = 0.
Q2: Determine the rational solutions for the equation: 4x4 – 6x3 – 26x2 + 12x – 9 = 0.
Q3: Solve the equation: 2x3 – 5x2 – x + 2 = 0 by identifying all possible rational solutions and testing them.
Q4: Find the rational solutions for the equation: x4 – 7x3 + 15x2 – 9x + 2 = 0.
Q5: Consider the equation: 5x3 – 8x2 – 14x + 12 = 0. Determine all the rational solutions and check if they satisfy the equation.
Summary – Rational Root Theorem
The Rational Root Theorem is a fundamental concept in algebra that helps identify possible rational roots of polynomial equations with integer coefficients. According to the theorem, for a rational number, expressed as p/q , to be a root of a polynomial equation, the numerator p must be a factor of the constant term at the end of the polynomial, and the denominator q must be a factor of the leading coefficient, which is the coefficient of the term with the highest degree. This theorem streamlines the process of finding rational solutions by providing a systematic way to test potential candidates. It’s particularly useful for solving polynomial equations efficiently, making it a staple in both educational settings and practical applications in mathematics.
FAQs on Rational Root Theorem
What is the Rational Root Theorem in Algebra?
The Rational Root Theorem is a fundamental concept in algebra that helps us identify potential rational solutions (in the form of fractions) to polynomial equations with whole number coefficients. It establishes conditions that these solutions must meet.
What are the Conditions of Rational Root Theorem?
There are two core conditions:
- The leading coefficient of the polynomial must be divisible by the denominator of the fraction representing the rational solution.
- The constant term of the polynomial must be divisible by the numerator of the fraction representing the rational solution.
How do you apply the Rational Root Theorem to find rational solutions?
To apply the theorem, you first identify the leading coefficient and the constant term of the polynomial equation. Then, you list the divisors of both coefficients. Potential rational solutions are fractions formed by pairing the divisors in the form p/q, where p is a divisor of the constant term, and q is a divisor of the leading coefficient.
What is the significance of Rational Root Theorem?
The theorem is essential in algebra as it provides a systematic approach to identifying rational solutions to polynomial equations. It aids mathematicians in solving various mathematical problems, including those in the physical sciences. It also forms the foundation for more advanced calculus and algebraic techniques.
Can the Rational Root Theorem guarantee that a rational solution exists for every polynomial equation?
No, the theorem does not guarantee that a rational solution will always exist. It provides potential rational solutions that need to be tested in the equation. In some cases, there may be no rational solutions, and other techniques or methods are needed.
How to Check Rational Solutions in a Polynomial Equation?
To check rational zeros, substitute each potential solution into the equation and calculate the result. If the result is equal to zero, the candidate is a valid rational solution. If the result is not zero, the candidate is not a rational solution.
Can the Rational Root Theorem be used for Equations with Irrational or Complex Solutions?
No, the theorem is specifically for identifying rational solutions. It doesn’t apply to equations with irrational or complex solutions.
What is Rational Root Equation?
The Rational Root Equation is given as x = p/q where p is the divisor of constant term and q is the divisor of the leading coefficient.
Are Rational Root Theorem, Rational Zeros Theroem and Rational Root Test same?
Yes the Rational Root Theorem, Rational Zeros Theorem and Rational Root Test all are same.
Share your thoughts in the comments
Please Login to comment...