Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ \left(x^{1 / 3}-y^{1 / 3}\right)\left(x^{2 / 3}+x^{1 / 3} y^{1 / 3}+y^{2 / 3}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to the polynomial \( x - y \).

Step 1: Understand the Problem

We are given an expression \[ (x^{1/3} - y^{1/3})(x^{2/3} + x^{1/3}y^{1/3} + y^{2/3}) \] that needs to be expressed as a polynomial. Note that this is a difference of cubes factorization in a disguised form. We will expand this expression step-by-step.

Step 2: Recall the Identity

Recognize that the given expression follows the identity for the difference of cubes: \ \[ a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2}) \]Here, let \( a = x^{1/3} \) and \( b = y^{1/3} \). Our objective is to apply this identity.

Step 3: Apply the Identity

Substitute \( a = x^{1/3} \) and \( b = y^{1/3} \) into the identity. We get:\[ (x^{1/3})^3 - (y^{1/3})^3 = (x^{1/3} - y^{1/3})(x^{2/3} + x^{1/3}y^{1/3} + y^{2/3}) \]Simplify the cubes:\[ x - y = (x^{1/3} - y^{1/3})(x^{2/3} + x^{1/3}y^{1/3} + y^{2/3}) \]

Step 4: Express as a Polynomial

Now we know that\[ (x^{1/3} - y^{1/3})(x^{2/3} + x^{1/3}y^{1/3} + y^{2/3}) = x - y \]Thus, the given expression is simply the polynomial \( x - y \).

Key Concepts

Polynomials

A polynomial is an algebraic expression that consists of variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents of variables. In simpler terms, polynomials are expressions like

• 3x^2 + 4x - 5
• 2a^3 + 3a - 8

Polynomials are named based on their degree, which is the highest power of the variable in the expression. For instance:

• A linear polynomial has a degree of 1, like 2x + 3.
• A quadratic polynomial has a degree of 2, like x^2 - 4x + 4.
• A cubic polynomial has a degree of 3, like x^3 - 2x^2 + x - 5.

The highest degree term is significant as it dictates the polynomial's shape and behavior as the variable's value changes. Polynomials are the foundation of algebra and form the building blocks for more advanced concepts in mathematics.
Understanding the characteristics of polynomials, such as their degree, is crucial because it helps identify the type of solutions associated with an equation, as seen when solving cubic equations.

Cubic Equations

Cubic equations are polynomials of degree three, having the general form\[ ax^3 + bx^2 + cx + d = 0 \] where \( a \), \( b \), \( c \), and \( d \) are constants with \( a eq 0 \). Solving them means finding the values of \( x \) that satisfy the equation.
Cubic equations can initially seem complex due to their degree, but there are methods to simplify them. Traditional approaches include:

• Factoring: Finding two or more simpler polynomials that multiply to give the cubic.
• Using the Rational Root Theorem: Helps in identifying possible rational roots.
• Applying the quadratic formula: When the cubic is factored into a quadratic and a linear factor.

For special identities like the difference of cubes, understanding these techniques can greatly ease the process of solving or simplifying equations. Comparing our given example,
\[(x^{1/3} - y^{1/3})(x^{2/3} + x^{1/3}y^{1/3} + y^{2/3})\] supposedly equaling \( x-y \), might initially seem surprising but proves simpler when considering its factored identity form derived from cubes.

Factorization

Factorization is a critical process in algebra, defined as breaking a complex expression into simpler terms or parts (factors) that, when multiplied together, recreate the original expression. This often simplifies solving and manipulation of algebraic expressions.
In algebra, some familiar factorization patterns involve:

• Difference of Squares: \( a^2 - b^2 = (a + b)(a - b) \)
• Difference of Cubes: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)
• Sum of Cubes: \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \)

The importance of recognizing these patterns lies in their power to simplify potentially daunting algebraic expressions, like the one in our original exercise. Given an expression like \[(x^{1/3} - y^{1/3})(x^{2/3} + x^{1/3}y^{1/3} + y^{2/3})\] the difference of cubes identity helps us factor and express it simply as the polynomial \( x - y \). Knowing when and how to apply these identities is key to mastering algebra.