Problem 33 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ x^{18}-y^{50} $$

Step-by-Step Solution

Answer

The factors are \((x^9 - y^{25})(x^9 + y^{25})\). Both are prime polynomials.

Step 1: Identify the Pattern

Recognize that the given expression is in the form of the difference of two powers, which can be expressed as \(a^n - b^m\). For the given polynomial, \(a = x^9\) and \(b = y^{25}\), resulting in the form \((x^9)^2 - (y^{25})^2\).

Step 2: Apply the Difference of Squares Formula

Use the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), to factor the expression. Substituting \(a = x^9\) and \(b = y^{25}\), the expression becomes: \( (x^9 - y^{25})(x^9 + y^{25}) \).

Step 3: Check for Prime Polynomials

Next, check each factor to see if they can be factored further. The factors \(x^9 - y^{25}\) and \(x^9 + y^{25}\) cannot be factored further using standard factoring techniques for polynomials, and both are considered prime polynomials.

Key Concepts

Difference of Squares

The difference of squares is a common algebraic expression. It appears in the form \(a^2 - b^2\). You can factor it using the formula \(a^2 - b^2 = (a - b)(a + b)\). This formula is useful for simplifying polynomials.
When you see \(a^2 - b^2\), think of it as finding two terms that, when multiplied together, produce the original expression.
For example, if you have \(x^{18} - y^{50}\), you can rewrite it using smaller powers: \(a = x^9\) and \(b = y^{25}\). This turns the original expression into \((x^9)^2 - (y^{25})^2\).
Using the difference of squares formula, you can now factor it as: \((x^9 - y^{25})(x^9 + y^{25})\). This makes complex expressions more manageable.

Prime Polynomials

A prime polynomial cannot be factored using standard algebraic methods. This means you cannot break it down further into simpler factors.
To determine if a polynomial is prime, you try to factor it. If it doesn’t work, the polynomial is prime.
In our example, \(x^9 - y^{25}\) and \(x^9 + y^{25}\) are both prime polynomials. We know this because no further factoring is possible using common techniques such as difference of squares or other factoring formulas.
Recognizing prime polynomials helps in knowing when you're done with factoring.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (such as +, -, *, and /).
Polynomials are a type of algebraic expression that include terms with variables raised to whole number exponents.
For example, \(x^{18} - y^{50}\) is an algebraic expression and specifically a polynomial. To factor polynomials, we use patterns and mathematical rules.
Factoring helps simplify algebraic expressions and solve equations. It breaks down complicated expressions into more manageable parts.
Learning different factoring methods allows us to handle a wide range of algebraic expressions effectively.