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Chapter 5: Problem 100

Factor completely. \(x^{-4}+2 x^{-5}+x^{-6}\)

Short Answer

Expert verified
\(x^{-6}(x + 1)^2\)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Look at the given expression: \(x^{-4}+2x^{-5}+x^{-6}\)Identify the smallest exponent of \(x\) in the terms, which is \(-6\). The GCF of the terms is \(x^{-6}\).
02

Factor out the GCF

Factor out \(x^{-6}\) from each term of the expression. \(x^{-4}+2x^{-5}+x^{-6} = x^{-6}(x^2 + 2x + 1)\)
03

Simplify the remaining expression

Simplify the expression inside the parentheses:\(x^2 + 2x + 1\)This expression can be factored further as a perfect square trinomial:\((x+1)^2\)
04

Write the final factored form

Combine the GCF and the simplified expression:\(x^{-6}(x + 1)^2\)

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Key Concepts

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

Greatest Common Factor (GCF)
In algebra, determining the greatest common factor (GCF) is crucial for simplifying expressions. The GCF is the largest factor that divides all terms in an expression. To identify the GCF, examine the coefficients and the variable exponents. In the expression \(x^{-4} + 2x^{-5} + x^{-6}\), the smallest exponent is \(-6\). Hence, the GCF is \(x^{-6}\). Factoring the GCF simplifies the expression and makes further manipulation easier.
Negative Exponents
Negative exponents can be tricky but understanding them is essential in algebra. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, \(x^{-a} = \frac{1}{x^a}\).When factoring expressions with negative exponents, remember to find the smallest exponent as it represents the GCF. Once factored out, negative exponents help to simplify complex algebraic expressions. For example, in \(x^{-4} + 2x^{-5} + x^{-6}\), factoring out \(x^{-6}\) changes the expression into \(x^{-6}(x^2 + 2x + 1)\).
Perfect Square Trinomial
A perfect square trinomial is an expression that results from squaring a binomial. It's of the form \((a + b)^2 = a^2 + 2ab + b^2\) or \((a - b)^2 = a^2 - 2ab + b^2\). Identifying a perfect square trinomial allows further factorization. For instance, in \(x^2 + 2x + 1\), we recognize it as a perfect square trinomial because it matches \((x + 1)^2 = x^2 + 2x + 1\). So, the expression inside the parentheses in \(x^{-6}(x^2 + 2x + 1)\) simplifies to \(x^{-6}(x + 1)^2\). This knowledge speeds up solving and simplifies algebraic expressions.

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