Question

Show that \(\sqrt{2}\) is algebraic.

Short answer

\(\sqrt{2}\) is an algebraic number because it is a root of the polynomial \(x^2 - 2 = 0\).

Step-by-step solution

Define Algebraic Numbers

An algebraic number is a number that is a root of a non-zero polynomial equation with rational coefficients. Our task is to demonstrate that \(\sqrt{2}\) satisfies this condition.

Identify a Suitable Polynomial

To show that \(\sqrt{2}\) is algebraic, we need to find a polynomial with rational coefficients that has \(\sqrt{2}\) as a root. Consider the polynomial equation \(x^2 - 2 = 0\).

Verify the Root

Substitute \(x = \sqrt{2}\) into the polynomial equation: \((\sqrt{2})^2 - 2 = 0\). Calculate \((\sqrt{2})^2 = 2\). Therefore, the equation becomes \(2 - 2 = 0\), which holds true.

Conclude the Argument

Since \(\sqrt{2}\) satisfies the polynomial \(x^2 - 2 = 0\) with rational coefficients (where the coefficients are 1, 0, and -2), \(\sqrt{2}\) is indeed an algebraic number.

Key concepts

Polynomial Equations

Polynomial equations are fundamental to algebra and describe relationships involving variables and constants.
A polynomial equation is expressed in the standard form: \[a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\]where each \(a_i\) is a coefficient and \(n\) is a non-negative integer denoting the power of the variable \(x\).
The coefficients can be rational numbers, integers, or even other types of numbers.

• Constant term: This is the term without the variable, represented as \(a_0\).
• Linear term: This involves the variable to the first power, represented as \(a_1x\).
• Quadratic term: Involves the variable squared, represented as \(a_2x^2\).

Understanding polynomial equations is crucial because they can help describe diverse phenomena, from physics to economics. To classify some number as algebraic, it must be a root of such an equation with rational coefficients.

Rational Coefficients

Rational coefficients are a key part of algebraic polynomials. Rational numbers are numbers that can be expressed as quotients of integers (e.g., \(\frac{1}{2}, \frac{-4}{3}, 5\)).
When a polynomial has rational coefficients, each coefficient in the expression is a rational number. For example, consider the polynomial:\[3x^3 - \frac{1}{2}x^2 + 5x - 4 = 0\]In this equation, all the coefficients (3, \(-\frac{1}{2}\), 5, and -4) are rational.
Rational coefficients ensure the equation maintains predictable properties and makes it possible to determine algebraic numbers easily.

• Rational coefficients link polynomials to real-world problems where fractions naturally occur.
• They ensure clarity and precision in mathematical expressions and solutions.

Whether you are solving for roots or graphing curves, using rational coefficients helps solidify understanding and prediction of outcomes.

Roots of Equations

Finding the roots of an equation is a central endeavor in algebra. A root of an equation is a solution that satisfies the equation, making it true when substituted for the variable.
For polynomial equations, roots are the values of \(x\) that satisfy:\[a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\]In simple terms, when you plug the root back into the equation, the left side equals zero.

• In the equation \(x^2 - 2 = 0\), \(\sqrt{2}\) is a root because substituting \(\sqrt{2}\) for \(x\) makes the equation true.
• Verifying roots is crucial to prove certain numbers are algebraic, as illustrated by showing \(\sqrt{2}\) is a root of a polynomial with rational coefficients.

Understanding roots helps in analyzing functions graphically and algebraically, aiding in solving real-world problems like optimizing outputs or calculating break-even points.