rational root theorem Archives – Mathemafia

Rational Root Theorem – Proof

Sunil Kumar — Sat, 15 Feb 2025 14:54:11 +0000

Rational Root Theorem

Statement: Given a polynomial

𝑃(𝑥)=𝑎𝑛𝑥𝑛+𝑎𝑛−1𝑥𝑛−1+𝑎𝑛−2𝑥𝑛−2+…….+𝑎1𝑥+𝑎0

with integral coefficients where a_n ≠ 0, a₀ ≠ 0. if P(x) has a rational root p/q where p and q are relatively prime integers, then p is a divisor of a₀ and q is a divisor of a_n.

Proof

𝑃(𝑥)=𝑎𝑛𝑥𝑛+𝑎𝑛−1𝑥𝑛−1+𝑎𝑛−2𝑥𝑛−2+…….+𝑎1𝑥+𝑎0𝑤ℎ𝑒𝑟𝑒 𝑎0,…,𝑎𝑛∈ℤ 𝑎𝑛𝑑 𝑎0,𝑎𝑛≠0.

Let p/q be a rational root of this equation where p and q are coprime integers and q ≠ 0.

⇒𝑃(𝑝𝑞)=0⇒𝑎𝑛(𝑝𝑞)𝑛+𝑎𝑛−1(𝑝𝑞)𝑛−1+𝑎𝑛−2(𝑝𝑞)𝑛−2+…….+𝑎1(𝑝𝑞)+𝑎0=0

Multiply both sides by qⁿ

𝑞𝑛×[𝑎𝑛(𝑝𝑞)𝑛+𝑎𝑛−1(𝑝𝑞)𝑛−1+𝑎𝑛−2(𝑝𝑞)𝑛−2+…….+𝑎2(𝑝𝑞)2+𝑎1(𝑝𝑞)+𝑎0]=𝑞𝑛×0𝑎𝑛(𝑝𝑞)𝑛𝑞𝑛+𝑎𝑛−1(𝑝𝑞)𝑛−1𝑞𝑛+𝑎𝑛−2(𝑝𝑞)𝑛−2𝑞𝑛+…….+𝑎2(𝑝𝑞)2𝑞𝑛+𝑎1(𝑝𝑞)𝑞𝑛+𝑎0𝑞𝑛=0𝑎𝑛𝑝𝑛+𝑎𝑛−1𝑝𝑛−1𝑞+𝑎𝑛−2𝑝𝑛−2𝑞2+…….+𝑎2𝑝2𝑞𝑛−2+𝑎1𝑝𝑞𝑛−1+𝑎0𝑞𝑛=0 …(1)

Imp: We multiplied by qⁿto get rid of terms in the denominator.

Video tutorial: Proof of Rational Root Theorem

Proof of Part 1: p divides a₀

Let’s write equation (1) again

𝑎𝑛𝑝𝑛+𝑎𝑛−1𝑝𝑛−1𝑞+𝑎𝑛−2𝑝𝑛−2𝑞2+…….+𝑎2𝑝2𝑞𝑛−2+𝑎1𝑝𝑞𝑛−1+𝑎0𝑞𝑛=0 …(1)

Take a₀qⁿ to the other side of the equation.

𝑎𝑛𝑝𝑛+𝑎𝑛−1𝑝𝑛−1𝑞+𝑎𝑛−2𝑝𝑛−2𝑞2+…….+𝑎2𝑝2𝑞𝑛−2+𝑎1𝑝𝑞𝑛−1= –𝑎0𝑞𝑛

Now p is common in every term on LHS. This implies:

𝑝[𝑎𝑛𝑝𝑛−1+𝑎𝑛−1𝑝𝑛−2𝑞+𝑎𝑛−2𝑝𝑛−3𝑞2+…….+𝑎2𝑝𝑞𝑛−2+𝑎1𝑞𝑛−1]= –𝑎0𝑞𝑛

Now p, q and a₀, a₁,….a_n,all are integers. Hence LHS is an integer.

⇒𝑝|𝐿𝐻𝑆⇒𝑝|𝑅𝐻𝑆⇒𝑝|𝑎0𝑞𝑛

As we know that p and q are coprime

⇒𝑝 does not divide 𝑞⇒𝑝 does not divide 𝑞𝑛

But because 𝑝|𝑎0𝑞𝑛⇒𝑝|𝑎0

Proof of Part 2: q divides a_n

Let’s go back to equation (1)

𝑎𝑛𝑝𝑛+𝑎𝑛−1𝑝𝑛−1𝑞+𝑎𝑛−2𝑝𝑛−2𝑞2+…….+𝑎2𝑝2𝑞𝑛−2+𝑎1𝑝𝑞𝑛−1+𝑎0𝑞𝑛=0 …(1)

Take a_npⁿ to the other side of the equation.

𝑎𝑛−1𝑝𝑛−1𝑞+𝑎𝑛−2𝑝𝑛−2𝑞2+…….+𝑎2𝑝2𝑞𝑛−2+𝑎1𝑝𝑞𝑛−1+𝑎0𝑞𝑛=−𝑎𝑛𝑝𝑛

Now q is common in every term on LHS. This implies:

𝑞[𝑎𝑛−1𝑝𝑛−1+𝑎𝑛−2𝑝𝑛−2𝑞+…….+𝑎2𝑝2𝑞𝑛−3+𝑎1𝑝𝑞𝑛−2+𝑎0𝑞𝑛−1]=−𝑎𝑛𝑝𝑛

Now p, q and a₀, a₁,….a_n,all are integers. Hence LHS is an integer.

⇒𝑞|𝐿𝐻𝑆⇒𝑞|𝑅𝐻𝑆⇒𝑞|𝑎𝑛𝑝𝑛

As we know that p and q are coprime

⇒𝑞 does not divide 𝑝⇒𝑞 does not divide 𝑝𝑛

But because 𝑞|𝑎𝑛𝑝𝑛⇒𝑞|𝑎𝑛

Hope you find this proof useful.

Let’s now use this theorem to find the roots of a polynomial of degree 3.

Example of Rational Root Theorem

𝑃(𝑥)=2𝑥3–3𝑥2–2𝑥+3

Find the roots of P(x) using Rational Root Theorem.

This equation is of the form

𝑃(𝑥)=𝑎3𝑥3+𝑎2𝑥2+𝑎1𝑥+𝑎0

⇒𝑎3=2 𝑎𝑛𝑑 𝑎0=3

As per rational root theorem, if this polynomial has any rational root, say p/q, then

⇒ 𝑝 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎0 , 𝑞 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎3𝑝 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 3 , 𝑞 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 2

Here are the possible values of p/q:

𝑝𝑞⇒𝑝𝑞=±1, ±3=±1, ±2=±11,±31,±12,±32=±1,±3,±12,±32

Now, if P(x) has any rational root, it has to be from the possible values of p/q written above. If P(x) = 0 for any of these values, that will be a rational root of the polynomial. The three values for which P(x) = 0 are 1, -1 and 3/2.

𝑃(𝑥)𝑃(1)𝑃(−1)𝑃(32)=2𝑥3–3𝑥2–2𝑥+3=2−3−2+3=0=−2−3+2+3=0=2×278−3×94−2×32+3=274–274−3+3=0

Rational Root Theorem is a very important theorem of Polynomials and is intended for the students learning higher mathematics and for those who are willing to dive deep into the core mathematics. For lower classes, like class 9 and class 10, this theorem comes handy when we have to find the roots of polynomials with degree 3 or more.

\(\begin{aligned} \end{aligned}\)

The post Rational Root Theorem – Proof appeared first on Mathemafia.

Rational root theorem

Sunil Syal — Mon, 28 Sep 2020 03:38:13 +0000

Rational root theorem: Given a polynomial P(x) = a_nxⁿ + a_n-1x^n-1 + …… + a₁x + a₀ with integral coefficients, a_n≠0. If has a rational root p/q with p & q relatively prime positive integers, then p is a divisor of a₀ and q is a divisor of a_n.

About & Prerequisite

This is a very important theorem about polynomials which helps finding the rational roots of high degree polynomial equations. This actually is a theorem for the higher classes (generally part of graduation classes) but I personally feel that it is not at all a difficult theorem to learn. More importantly, a good theorem to learn in case you are dealing with rational roots of high degree polynomials.

You must be aware of basics of polynomials before you watch this tutorial.

Basics of polynomials

Chapter 2 – Polynomials

Here you will find complete solution of all questions, concepts, theorems and examples of class 9 chapter 2 (Polynomials) of class 9, NCERT textbook.

There are few very important concepts that students must learn to get a good understanding of polynomials:

All above mentioned tutorials are included in the list displayed below for class 9 chapter 2 (Polynomials). You are strongly advised to watch all these videos thoroughly and do not miss any of these if you really want to get good hold on polynomials.

Click a thumbnail to watch the tutorial.

Watch complete chapter on YouTube

Introduction to Polynomials | Introduction to Algebra | Class 9, Chapter 2, NCERT Maths

Class 9 | Chapter 2 | Polynomials Example 1 | Finding Degree | NCERT Maths

Question 1 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 2 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 3 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 4 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 5 Exercise 2.1 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Class 9 | Chapter 2 | Polynomials Example 2 explained | NCERT Maths

Class 9 | Chapter 2 | Polynomials Example 3 explained | NCERT Maths

Class 9 | Chapter 2 | Polynomials Example 4 explained | NCERT Maths

Class 9 | Chapter 2 | Polynomials Example 5 explained | NCERT Maths

Question 1 Exercise 2.2 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 2 Exercise 2.2 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 3 Exercise 2.2 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 4 Exercise 2.2 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

FACTOR THEOREM Proof Explained | Class 9 | Chapter 2 | Polynomials, 2.5 Factorisation of Polynomials

Example 6 Chapter 2 Class 9 | Polynomials, Factor Theorem Questions | NCERT Maths

Example 7 Chapter 2 Class 9 | Polynomials, Factor Theorem Questions | NCERT Maths

Factorisation of Polynomials by SPLITTING THE MIDDLE TERM Explained| Class 9 |Chapter 2| NCERT Maths

Example 8 Chapter 2 Class 9 | Polynomials | 1St Solution | Splitting Middle Term Questions | Maths

Example 8 Chapter 2 Class 9 | Polynomials | 2nd Solution | Factor Theorem Questions |NCERT Maths

Example 9 Chapter 2 Class 9 | Polynomials | Factorise using Factor Theorem Questions |NCERT Maths

Example 10 Chapter 2 Class 9 | Polynomials | Factorise cubic or degree 3 polynomials | NCERT Maths

Question 1 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 2 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 3 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 4 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 5 Exercise 2.3 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

EASY TIPS & TRICKS 🔥 TO LEARN ALGEBRAIC IDENTITIES | Class 9 | Chapter 2 | Polynomials | NCERT Maths

Example 11 Chapter 2 Class 9 | Polynomials | Product of polynomials using algebraic identities

Example 12 Chapter 2 Class 9 | Polynomials| Example 17 |Product of big numbers without multiplying

Example 13 Chapter 2 Class 9 | Polynomials | Factorise polynomials using algebra identities | NCERT

Example 14 Chapter 2 Class 9 | Polynomials | Expand polynomials using algebraic identities | NCERT

Example 15 Chapter 2 Class 9 | Polynomials | How to expand polynomials using algebraic formulae |

Example 16 Chapter 2 Class 9 | Polynomials | Factorise polynomials using algebraic identities

Example 17 Chapter 2 Class 9 | Polynomials | How to expand polynomials using algebraic formulae

Example 18 Chapter 2 Class 9 | Polynomials | Example 23 | Cubes of big numbers without multiplying

Example 19 Chapter 2 Class 9 | Polynomials |Factorise cubic polynomials using algebraic identities

Example 20 Chapter 2 Class 9 | Polynomials | Factorise cubic polynomials using algebraic identities

Question 1 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 2 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 3 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 4 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 5 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 6 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 7 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 8 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 9 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 10 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Question 11 Exercise 2.4 Class 9 | Chapter 2 (Polynomials) | NCERT Maths | Solution

Class 9 Chapter 3 | Coordinate Geometry

SHARING IS CARING. 😎 If you really like these videos, please share these with your friends and classmates. I have created these videos with a lot of efforts and are absolutely free for everyone. I alone cannot reach out to every student but you can help a lot. 🙏 🙏 🙏 I will really appreciate if you share it with someone. You can use WhatsApp, SnapChat, Twitter, Facebook, Messenger or any other social platform from the icons displayed below. Keep watching and sharing Mathemafia!!! 😎 Cheers!!!

rational root theorem Archives – Mathemafia

Rational Root Theorem – Proof