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Chapter 5: Problem 102

Factor completely. Assume variables used as exponents represent positive integers. $$x^{2 n}-9$$

Short Answer

Expert verified
(x^n - 3)(x^n + 3)

Step by step solution

01

Identify the type of polynomial

Recognize that the given expression, \( x^{2n} - 9 \), is a difference of squares. This type of polynomial can generally be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \).
02

Express as a difference of squares

Rewrite \( x^{2n} \) and \( 9 \) as squared terms: \( x^{2n} = (x^n)^2 \) and \( 9 = 3^2 \). The expression can now be written as \( (x^n)^2 - 3^2 \).
03

Apply the difference of squares formula

Use the difference of squares formula: \( (x^n)^2 - 3^2 = (x^n - 3)(x^n + 3) \).
04

Simplify and verify

The factored form of the given polynomial is \( (x^n - 3)(x^n + 3) \). Verify by expanding the factors to ensure correctness: \[(x^n - 3)(x^n + 3) = x^{2n} + x^n \times 3 - x^n \times 3 - 9 = x^{2n} - 9.\] This confirms that the factoring is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The 'difference of squares' is a special technique in factoring polynomials. It applies when you have two squared terms separated by a subtraction sign, like this:
  • \(a^2 - b^2\)
This can be factored using the formula:
  • \(a^2 - b^2 = (a - b)(a + b)\)
The reasoning behind this formula is that when you expand \((a - b)(a + b)\) using the distributive property (FOIL method), you'll end up with
  • \(a^2 + ab - ab - b^2\)
The middle terms cancel each other out, leaving just
  • \(a^2 - b^2\).
This is why the difference of squares formula is so useful and can quickly simplify expressions.
Factoring
Factoring is breaking down a polynomial into simpler 'factors' that when multiplied together give the original polynomial. In our example, the polynomial is
  • \(x^{2n} - 9\).
We've identified that this polynomial is a difference of squares. To factor it, we follow these steps:
  • Rewrite each term as a square: \(x^{2n} = (x^n)^2\); \(9 = 3^2\).
  • Use the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\).
  • Apply the formula: \((x^n)^2 - 3^2 = (x^n - 3)(x^n + 3)\).
Factoring breaks down a complex polynomial into simpler parts, which helps in solving equations or simplifying expressions further.
Polynomial
A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. For example,
  • \(x^{2n} - 9\).
Polynomials come in various forms:
  • Monomials: Single term (\(5x\)).
  • Binomials: Two terms (\(x+3\)).
  • Trinomials: Three terms (\(x^2 + 5x + 6\)).
The degree of a polynomial is determined by the highest power of the variable. In our example, the degree is
  • \(2n\) since the highest power of the variable is \(x^{2n}\).
Understanding the structure of polynomials is essential because it helps in recognizing patterns and applying appropriate factoring techniques.

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