Difference between revisions of "Rational root theorem"

Latest revision as of 12:07, 20 February 2024

In algebra, the rational root theorem states that given an integer polynomial $P(x)$ with leading coefficient $a_n$ and constant term $a_0$ , if $P(x)$ has a rational root $r = p/q$ in lowest terms, then $p|a_0$ and $q|a_n$ .

This theorem is most often used to guess the roots of polynomials. It sees widespread usage in introductory and intermediate mathematics competitions.

Proof

Let $\frac{p}{q}$ be a rational root of $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0$ , where every $a_r$ is an integer; we wish to show that $p|a_0$ and $q|a_n$ . Since $\frac{p}{q}$ is a root of $P(x)$ , $0 = a_n \left(\frac{p}{q}\right)^n + a_{n-1} \left(\frac{p}{q}\right)^{n-1} + \cdots + a_1 \left(\frac{p}{q}\right) + a_0.$ Multiplying by $q^n$ yields $0 = a_n p^n + a_{n-1} p^{n-1} q + \cdots + a_1 p * q^{n-1} + a_0 q^n.$ Using modular arithmetic modulo $p$ , we have $a_0 q^n \equiv 0\pmod p$ , which implies that $p | a_0 q^n$ . Because we've defined $p$ and $q$ to be relatively prime, $\gcd(q^n, p) = 1$ , which implies $p | a_0$ by Euclid's lemma. Via similar logic in modulo $q$ , $q|a_n$ , as required. $\square$

Intro to Rational Roots theorem: https://www.youtube.com/shorts/wKpmfnyKeeM

Examples

Here are some problems with solutions that utilize the rational root theorem.

Example 1

Find all rational roots of the polynomial $x^4-x^3-x^2+x+57$ .

Solution: The polynomial has leading coefficient $1$ and constant term $3 \cdot 19$ , so the rational root theorem guarantees that the only possible rational roots are $-57$ , $-19$ , $-3$ , $-1$ , $1$ , $3$ , $19$ , and $57$ . After testing every number, we find that none of these are roots of the polynomial; thus, the polynomial has no rational roots. $\square$

Example 2

Factor the polynomial $x^3-5x^2+2x+8$ .

Solution: After testing the divisors of 8, we find that it has roots $-1$ , $2$ , and $4$ . Then because it has leading coefficient $1$ , the factor theorem tells us that it has the factorization $(x-4)(x-2)(x+1), x={-1, 2, 4}$ . $\square$

Example 3

Using the rational root theorem, prove that $\sqrt{2}$ is irrational.

Solution: The polynomial $x^2 - 2$ has roots $\pm \sqrt{2}$ . The rational root theorem guarantees that the only possible rational roots of this polynomial are $-2, -1, 1$ , and $2$ . Testing these, we find that none are roots of the polynomial, and so it has no rational roots. Then because $\sqrt{2}$ is a root of the polynomial, it cannot be a rational number. $\square$

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-In [[algebra]], the '''rational root theorem''' states that given an integer [[polynomial]] <math>P(x)</math> with leading coefficient <math>a_n</math> and constant term <math>a_0</math>, if <math>P(x)</math> has a rational root <math>r = \frac pq</math> in lowest terms, then <math>p|a_0</math> and <math>q|a_n</math>.
+In [[algebra]], the '''rational root theorem''' states that given an integer [[polynomial]] <math>P(x)</math> with leading coefficient <math>a_n</math> and constant term <math>a_0</math>, if <math>P(x)</math> has a rational root <math>r = p/q</math> in lowest terms, then <math>p|a_0</math> and <math>q|a_n</math>.
-This theorem helps to find the "nice" roots of a polynomial significantly, aiding to how it limits the possible rational roots to a finite number.
+This theorem is most often used to guess the roots of polynomials. It sees widespread usage in introductory and intermediate mathematics competitions.
 == Proof ==
 Let <math>\frac{p}{q}</math> be a rational root of <math>P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0</math>, where every <math>a_r</math> is an integer; we wish to show that <math>p|a_0</math> and <math>q|a_n</math>. Since <math>\frac{p}{q}</math> is a root of <math>P(x)</math>, <cmath>0 = a_n \left(\frac{p}{q}\right)^n + a_{n-1} \left(\frac{p}{q}\right)^{n-1} + \cdots + a_1 \left(\frac{p}{q}\right) + a_0.</cmath> Multiplying by <math>q^n</math> yields <cmath>0 = a_n p^n + a_{n-1} p^{n-1} q + \cdots + a_1 p * q^{n-1} + a_0 q^n.</cmath> Using [[modular arithmetic]] modulo <math>p</math>, we have <math>a_0 q^n \equiv 0\pmod p</math>, which implies that <math>p | a_0 q^n</math>. Because we've defined <math>p</math> and <math>q</math> to be relatively prime, <math>\gcd(q^n, p) = 1</math>, which implies <math>p | a_0</math> by [[Euclid's lemma]]. Via similar logic in modulo <math>q</math>, <math>q|a_n</math>, as required. <math>\square</math>
-== Problems ==
-Here are some problems that are cracked by the rational root theorem.
+Intro to Rational Roots theorem: https://www.youtube.com/shorts/wKpmfnyKeeM
-=== Problem 1 ===
+== Examples ==
+Here are some problems with solutions that utilize the rational root theorem.
+=== Example 1 ===
 ''Find all rational roots of the polynomial <math>x^4-x^3-x^2+x+57</math>.
 '''Solution''': The polynomial has leading coefficient <math>1</math> and constant term <math>3 \cdot 19</math>, so the rational root theorem guarantees that the only possible rational roots are <math>-57</math>, <math>-19</math>, <math>-3</math>, <math>-1</math>, <math>1</math>, <math>3</math>, <math>19</math>, and <math>57</math>. After testing every number, we find that none of these are roots of the polynomial; thus, the polynomial has no rational roots. <math>\square</math>
-=== Problem 2 ===
+=== Example 2 ===
 ''Factor the polynomial <math>x^3-5x^2+2x+8</math>.''
-'''Solution''': After testing the divisors of 8, we find that it has roots <math>-1</math>, <math>2</math>, and <math>4</math>. Then because it has leading coefficient <math>1</math>, the [[factor theorem]] tells us that it has the factorization <math>(x-4)(x-2)(x+1)</math>. <math>\square</math>
+'''Solution''': After testing the divisors of 8, we find that it has roots <math>-1</math>, <math>2</math>, and <math>4</math>. Then because it has leading coefficient <math>1</math>, the [[factor theorem]] tells us that it has the factorization <math>(x-4)(x-2)(x+1), x={-1, 2, 4}</math>. <math>\square</math>
-=== Problem 3 ===
+=== Example 3 ===
 ''Using the rational root theorem, prove that <math>\sqrt{2}</math> is irrational.''
-'''Solution''': The polynomial <math>x^2 - 2</math> has roots <math>-\sqrt{2}</math> and <math>\sqrt{2}</math>. The rational root theorem garuntees that the only possible rational roots of this polynomial are <math>-2, -1, 1</math>, and <math>2</math>. Testing these, we find that none are roots of the polynomial, and so it has no rational roots. Then because <math>\sqrt{2}</math> is a root of the polynomial, it cannot be a rational number. <math>\square</math>
+'''Solution''': The polynomial <math>x^2 - 2</math> has roots <math>\pm \sqrt{2}</math>. The rational root theorem guarantees that the only possible rational roots of this polynomial are <math>-2, -1, 1</math>, and <math>2</math>. Testing these, we find that none are roots of the polynomial, and so it has no rational roots. Then because <math>\sqrt{2}</math> is a root of the polynomial, it cannot be a rational number. <math>\square</math>
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