Problem 43
Question
Use the rational root theorem to argue that \(\sqrt{2}\) is not a rational number. [Hint: The solutions of \(x^{2}-2=0\) are \(\pm \sqrt{2}\).]
Step-by-Step Solution
Verified Answer
\( \sqrt{2} \) is irrational because there are no rational roots for \( x^2 - 2 = 0 \).
1Step 1: Understand the problem
The exercise asks us to use the Rational Root Theorem to show that \( \sqrt{2} \) is not a rational number by considering the polynomial \( x^2 - 2 = 0 \). We need to show that \( \sqrt{2} \) cannot be expressed as a fraction of two integers \( \frac{p}{q} \).
2Step 2: Apply the Rational Root Theorem
The Rational Root Theorem states that if a polynomial with integer coefficients has a rational root \( \frac{p}{q} \), where \( p \) and \( q \) are integers with \( \gcd(p, q) = 1 \), then \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
3Step 3: Identify the coefficients
For the polynomial \( x^2 - 2 = 0 \), the constant term is \(-2\) and the leading coefficient is \(1\). According to the Rational Root Theorem, any rational root \( \frac{p}{q} \) must have \( p \) as a factor of \(-2\) and \( q \) as a factor of \(1\).
4Step 4: List possible rational roots
The possible values for \( p \) are \( \pm 1, \pm 2 \). Since \( q \) must divide \(1\), \( q = \pm 1 \). Thus, the possible rational roots are \( \pm 1, \pm 2 \).
5Step 5: Check possible rational roots
Test each possible rational root in the equation \( x^2 - 2 = 0 \). * For \( x = 1 \), \( 1^2 - 2 = -1 eq 0 \) * For \( x = -1 \), \( (-1)^2 - 2 = -1 eq 0 \) * For \( x = 2 \), \( 2^2 - 2 = 2 eq 0 \) * For \( x = -2 \), \( (-2)^2 - 2 = 2 eq 0 \) None of these values satisfy the equation, meaning there are no rational roots.
6Step 6: Conclude that \( \sqrt{2} \) is irrational
Since \( \sqrt{2} \) is a root of \( x^2 - 2 = 0 \) and there are no rational roots, \( \sqrt{2} \) cannot be a rational number.
Key Concepts
Irrational NumbersPolynomial EquationsInteger Coefficients
Irrational Numbers
Irrational numbers are real numbers that cannot be expressed as a simple fraction of two integers. They are numbers that, when written in decimal form, have non-terminating and non-repeating patterns. For example, numbers such as \( \pi \) (pi) and \( \sqrt{2} \) (the square root of 2) are irrational.
The exercise asks us to demonstrate that \( \sqrt{2} \) is irrational. This means \( \sqrt{2} \) cannot be written as \( \frac{p}{q} \), where \( p \) and \( q \) are integers with no common divisors other than 1. Traditionally, proving an irrational number involves contradiction: assuming the contrary and arriving at impossible conclusions. In this case, using the Rational Root Theorem does the trick and saves us from manually testing infinite possibilities.
The exercise asks us to demonstrate that \( \sqrt{2} \) is irrational. This means \( \sqrt{2} \) cannot be written as \( \frac{p}{q} \), where \( p \) and \( q \) are integers with no common divisors other than 1. Traditionally, proving an irrational number involves contradiction: assuming the contrary and arriving at impossible conclusions. In this case, using the Rational Root Theorem does the trick and saves us from manually testing infinite possibilities.
Polynomial Equations
A polynomial equation is a mathematical expression consisting of variables and coefficients, structured in a specific format. It combines terms using only addition, subtraction, and multiplication. Think of it as a string of numbers and variables like \( x^2 - 2 = 0 \).
In polynomial equations, the highest power of the variable dictates its degree, which can indicate the number of potential solutions or roots. A quadratic equation, like \( x^2 - 2 = 0 \), is second-degree because it contains \( x^2 \). This specific polynomial brings us to the roots \( \pm \sqrt{2} \). These roots imply that the equation describes a shape that crosses the x-axis at these points.
This exercise utilized a steady sequence to solve polynomial equations, ultimately checking if any rational solutions fit. Beyond theoretical exercises like this one, quadratic equations appear throughout STEM (Science, Technology, Engineering, and Mathematics) subjects in various forms, allowing us to solve real-world problems.
In polynomial equations, the highest power of the variable dictates its degree, which can indicate the number of potential solutions or roots. A quadratic equation, like \( x^2 - 2 = 0 \), is second-degree because it contains \( x^2 \). This specific polynomial brings us to the roots \( \pm \sqrt{2} \). These roots imply that the equation describes a shape that crosses the x-axis at these points.
This exercise utilized a steady sequence to solve polynomial equations, ultimately checking if any rational solutions fit. Beyond theoretical exercises like this one, quadratic equations appear throughout STEM (Science, Technology, Engineering, and Mathematics) subjects in various forms, allowing us to solve real-world problems.
Integer Coefficients
In mathematics, an integer is a whole number that can be positive, negative, or zero. Knowing what makes integer coefficients distinct clarifies the rules of certain mathematical principles, like the Rational Root Theorem.
When working with polynomial equations with integer coefficients, each term's coefficient is a straightforward whole number. This characteristic becomes crucial when determining possible rational roots because any rational root must satisfy specific conditions related to these coefficients.
For the polynomial \( x^2 - 2 \), we have integer coefficients of 1 (for \( x^2 \)) and -2 (the constant term). Using integer coefficients, our steps to applying the Rational Root Theorem become precise because our possibilities always tie back to these concrete whole numbers. This also illustrates integers' importance in building algebraic structures and simplifying complex problems.
When working with polynomial equations with integer coefficients, each term's coefficient is a straightforward whole number. This characteristic becomes crucial when determining possible rational roots because any rational root must satisfy specific conditions related to these coefficients.
For the polynomial \( x^2 - 2 \), we have integer coefficients of 1 (for \( x^2 \)) and -2 (the constant term). Using integer coefficients, our steps to applying the Rational Root Theorem become precise because our possibilities always tie back to these concrete whole numbers. This also illustrates integers' importance in building algebraic structures and simplifying complex problems.
Other exercises in this chapter
Problem 42
For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x+3, \quad f(x
View solution Problem 42
Use synthetic division to determine the quotient and remainder for each problem. $$ \left(4 x^{4}-5 x^{2}+1\right) \div\left(x-\frac{1}{2}\right) $$
View solution Problem 43
For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x+5, \quad f(x
View solution Problem 43
How would you give a general description of what is accomplished with synthetic division to someone who had just completed an elementary algebra course?
View solution