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Chapter 5: Problem 19

Multiply. $$\left(p-\frac{1}{4}\right)\left(p+\frac{1}{4}\right)$$

Short Answer

Expert verified
The product is \(p^2 - \frac{1}{16}\).

Step by step solution

01

Recognize the pattern

Notice that the given expression has the form \((a - b)(a + b)\). This is a difference of squares formula, which states \((a - b)(a + b) = a^2 - b^2\).
02

Identify the terms

In the given expression \(p - \frac{1}{4}\) and \(p + \frac{1}{4}\), identify \(a = p\) and \(b = \frac{1}{4}\).
03

Apply the difference of squares formula

Use the formula \((a - b)(a + b) = a^2 - b^2\). Here, it will be \((p - \frac{1}{4})(p + \frac{1}{4}) = p^2 - (\frac{1}{4})^2\).
04

Simplify

Finally, simplify \(p^2 - (\frac{1}{4})^2\). \((\frac{1}{4})^2\) is \frac{1}{16}\. So, the expression becomes \p^2 - \frac{1}{16}\.

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Key Concepts

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

algebraic identities
In algebra, identities are equations that are true for all values of the variables involved. One very useful identity is the difference of squares identity. This identity states that for any numbers or expressions \(a\) and \(b\): \((a - b)(a + b) = a^2 - b^2\). Recognizing when to use this identity can simplify complex multiplication problems.

In the given exercise, \((p - \frac{1}{4})(p + \frac{1}{4})\), we can identify that it fits the pattern of the difference of squares. Here, \(a = p\) and \(b = \frac{1}{4}\). By recognizing this, we can use the identity to simplify our work significantly.
polynomial multiplication
Polynomial multiplication involves distributing each term in one polynomial to every term in the other polynomial. Without the difference of squares identity, you would have to use the distributive property to multiply each term:

  • First, distribute \(p\): \( p \times (p + \frac{1}{4}) = p^2 + \frac{p}{4}\)
  • Then, distribute \(-\frac{1}{4}\): \( -\frac{1}{4} \times (p + \frac{1}{4}) = -\frac{p}{4} - \frac{1}{16}\)
  • Combine like terms: \(p^2 + \frac{p}{4} - \frac{p}{4} - \frac{1}{16} = p^2 - \frac{1}{16}\)

Using the identity saves us from these steps by immediately giving us \(p^2 - (\frac{1}{4})^2\). This demonstrates how algebraic identities simplify polynomial multiplication.
simplifying expressions
Simplifying expressions means making them as simple as possible. This often involves combining like terms, reducing fractions, and using identities like the difference of squares. Let's simplify \(p^2 - (\frac{1}{4})^2\).

First, find the value of \((\frac{1}{4})^2\):

\( \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\).

So \(p^2 - (\frac{1}{4})^2\) simplifies to \(p^2 - \frac{1}{16}\). This makes the expression easier to interpret and work with.

Understanding the steps to simplify expressions is crucial for solving algebra problems efficiently. When you recognize patterns and apply identities, solving and simplifying expressions becomes more straightforward.

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Most popular questions from this chapter

To prepare for Section 5.3, review combining like terms and evaluating expressions (Sections 1.6 and 1.8). Combine like terms. $$ 9 x+2 y-x-2 y $$

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Simplify. $$ 5^{0}-5^{-1} $$
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