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Chapter 5: Problem 55

Factor completely. \(\frac{1}{100}-x^{2}\)

Short Answer

Expert verified
\[ \frac{1}{100} - x^2 = \bigg(\frac{1}{10} - x \bigg)\bigg(\frac{1}{10} + x \bigg) \]

Step by step solution

01

Identify the Expression

Recognize that the given expression is \(\frac{1}{100} - x^2\). This is a difference of squares.
02

Rewrite as Difference of Squares

Rewrite \(\frac{1}{100}\) as \(\frac{1^2}{10^2}\), so the expression becomes \(\frac{1^2}{10^2}-x^2\).
03

Factor the Difference of Squares

Use the difference of squares formula \(a^2 - b^2 = (a-b)(a+b) \). Here, \ a = \frac{1}{10} \ and \ b = x \, so the factored form is \( (\frac{1}{10} - x)(\frac{1}{10} + x) \).

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Key Concepts

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

factoring polynomials
Factoring polynomials is a key skill in algebra. It involves breaking down a polynomial into simpler components that, when multiplied together, give you the original polynomial. This is similar to how numbers can be factored into primes.

In our example, we started with a polynomial \(\frac{1}{100} - x^2\). Factoring makes it easier to solve equations and understand properties of the expression. Let's dig into the exact steps used.
difference of squares
The difference of squares is a specific type of polynomial. It has the form \[a^2 - b^2 = (a-b)(a+b)\]. This formula is very useful for factoring because it reduces a complicated expression into a product of two binomials.

In our exercise, \[a = \frac{1}{10}\] and \[b = x\]. So when you rewrite \[ \frac{1}{100} \] as \[ \frac{1^2}{10^2} \], the expression fits the difference of squares form. Applying the formula, we get \[ (\frac{1}{10} - x)(\frac{1}{10} + x) \].
algebraic expressions
Algebraic expressions are combinations of numbers, variables, and operations. In our exercise, \(\frac{1}{100} - x^2\) is an algebraic expression made of a fraction and a variable squared. The goal of exercises like these is to simplify the expression as much as possible.

Understanding each part of the expression and how to manipulate it helps in solving more complex algebraic problems. Simplifying \[ \frac{1}{100} \] as \[ \frac{1^2}{10^2} \] and recognizing it as a difference of squares shows how fundamental skills in algebra build upon one another.

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Most popular questions from this chapter

Factor completely. $$ 28 a^{3}-25 a^{2} b c+3 a b^{2} c^{2} $$

Suppose that you are given a detailed graph of \(y=p(x),\) where \(p(x)\) is some polynomial in \(x\) How could the graph be used to help solve the equation \(p(x)=0 ?\)

Solve. If \(f(x)=2 x^{3}-5 x\) and \(g(x)=10 x-7 x^{2},\) find all \(x\) -values for which \(f(x)=g(x)\)

Find the domain of the function \(f\) given by each of the following. $$f(x)=\frac{x-5}{9 x-18 x^{2}}$$

Factor completely. $$ -6 x^{4}+8 x^{3}-12 x $$
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