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Chapter 6: Problem 84

Apply the special factoring nules of this section to factor each binomial or trinomial. $$ q^{2}-\frac{1}{4} $$

Short Answer

Expert verified
(q + 1/2)(q - 1/2)

Step by step solution

01

Recognize the difference of squares

The given expression is a binomial: \( q^2 - \frac{1}{4} \). Notice that both terms are perfect squares: \( q^2 \) is the square of \( q \), and \( \frac{1}{4} \) is the square of \( \frac{1}{2} \).
02

Apply the difference of squares formula

The difference of squares formula is \( a^2 - b^2 = (a + b)(a - b) \). Here, \( a = q \) and \( b = \frac{1}{2} \).
03

Factor the expression

Using the formula, factor \( q^2 - \frac{1}{4} \) into \( (q + \frac{1}{2})(q - \frac{1}{2}) \).

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Key Concepts

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

difference of squares
In algebra, the difference of squares is a specific quadratic form. When you have a binomial of the form Example: Note: The terms in the binomial must be perfect squares. For instance, in our example, 2. Subtract the second square term. Here it is . This makes it easier and faster to factor certain binomials.
factoring
Factoring is the process of breaking down an expression into a product of simpler expressions. When factoring a binomial like the one in our problem, it's important to:
  • Identify the factor with the same sign but different terms.
  • Here: Factor out the common term which results in 2.
    • Factor the expression until no further simplification is possible.
perfect squares
Perfect squares are numbers that can be written as the square of an integer or a rational number. To identify perfect squares in a binomial, you should: For example: Note: Not every quadratic binomial is a difference of squares.

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Most popular questions from this chapter

Factor each polynomial. $$ x^{9}+5 x^{8} w-24 x^{7} w^{2} $$

Factor each trinomial completely. $$ 4 k^{3}-4 k^{2}+9 k $$

Factor each polynomial completely. $$ 3 r-3 k+3 r^{2}-3 k^{2} $$

Factor each trinomial completely. $$ 16 x^{2}-40 x+25 $$

Factor by grouping. \(16 m^{3}-4 m^{2} p^{2}-4 m p+p^{3}\)
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