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Chapter 5: Problem 96

Factor. Assume that variables used as exponents represent positive integers. $$ x^{4 n}-625 $$

Short Answer

Expert verified
The expression factors into \((x^n - 5)(x^n + 5)(x^{2n} + 25)\).

Step by step solution

01

Identify the Structure of the Expression

The given expression is \(x^{4n} - 625\). This is a difference of squares structure because \(625\) is a perfect square, and can be written as \(25^2\).
02

Apply the Difference of Squares Formula

The formula for the difference of squares is \(a^2 - b^2 = (a - b)(a + b)\). In this case, take \(a = x^{2n}\) and \(b = 25\).
03

Factor Using the Formula

Substitute \(a\) and \(b\) into the difference of squares formula to factor the expression: \(x^{4n} - 625 = (x^{2n} - 25)(x^{2n} + 25)\).
04

Identify Potential Further Factoring

Check each factor to see if they can be factored further. Notice that \(x^{2n} - 25\) is another difference of squares because \(25\) is \(5^2\).
05

Factor Again Using the Difference of Squares

Apply the difference of squares formula again to \(x^{2n} - 25\): \(x^{2n} - 25 = (x^n - 5)(x^n + 5)\).
06

Present the Fully Factored Form

Combine all the factored terms: \(x^{4n} - 625 = (x^n - 5)(x^n + 5)(x^{2n} + 25)\).

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Key Concepts

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

Difference of Squares
When you come across expressions like \(x^{4n} - 625\), you might notice that it fits a special pattern known as the "difference of squares." This pattern is characterized by the structure \(a^2 - b^2\), where both \(a\) and \(b\) are squared terms. In our example:
  • The first term \(x^{4n}\) can be seen as \(a^2\) when rewritten as \( (x^{2n})^2\).
  • The second term \(625\) is actually \(25^2\).
When we apply the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\), we can factor the expression easily. This means taking the square root of both terms, \(a = x^{2n}\) and \(b = 25\), and then writing it as \( (x^{2n} - 25)(x^{2n} + 25)\).

However, the story doesn't end there. The expression \(x^{2n} - 25\) is itself a difference of squares, allowing for further factoring.
Exponents
Exponents can initially seem a little tricky, but they're important for simplifying and factoring expressions like \(x^{4n} - 625\). They indicate how many times a number, termed the base, is multiplied by itself. Let's delve further into the example given:
  • \(x^{4n}\) means \(x\) is multiplied by itself \(4n\) times. Think of \(4n \) simply as a number counting the multiplications.
  • Similarly, in the intermediate step, \(x^{2n}\) indicates that \(x\) is multiplied by itself \(2n\) times.
Understanding how to manage exponents is crucial.
  • When you square an expression like \(x^{2n}\), you multiply the exponent by 2, resulting in \(x^{4n}\).
  • Learning how expressions break down or get simplified due to exponents is key for smooth problem-solving.
Working through each factor stepwise requires precise handling of exponents to guarantee complete factoring.
Perfect Squares
A perfect square is a number that can be expressed as the square of an integer. Recognizing perfect squares in expressions is a helpful skill in factoring.In our example, \(625\) is a perfect square because it equals \(25^2\). This recognition allows the difference of squares strategy to be employed.
  • To determine if a number is a perfect square, simply check if its square root results in a clean integer. For \(625\), its square root is \(25\), which confirms it's a perfect square.
  • Another perfect square noticed earlier in factoring was \(x^{2n}\), which makes \(x^{4n}\) as the square of \(x^{2n}\).
When you factor such expressions, understanding perfect squares is essential because they point directly to the underlying patterns that make expressions easier to break down. By breaking associations with perfect squares, complex polynomials can often be made simpler.

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Most popular questions from this chapter

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}+4 x-5 $$

Factor each polynomial completely. See Examples 1 through 12. $$ 12 x^{3}+x^{2}-x $$

Recall that a graphing calculator may be used to check addition, subtraction, and multiplication of polynomials. In the same manner, a graphing calculator may be used to check factoring of polynomials in one variable. For example, to see that $$ 2 x^{3}-9 x^{2}-5 x=x(2 x+1)(x-5) $$ graph \(\mathrm{Y}_{1}=2 x^{3}-9 x^{2}-5 x\) and \(\mathrm{Y}_{2}=x(2 x+1)(x-5) .\) Then trace along both graphs to see that they coincide. Factor the following and use this method to check your results. $$ -6 x^{4}+10 x^{3}-4 x^{2} $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{4 n}-16 $$

Use substitution to factor each polynomial completely. See Examples 11 and 12. $$ (3 x-1)^{2}+5(3 x-1)+6 $$
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