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Chapter 5: Problem 58

Factor completely. \((p+q)^{2}-64\)

Short Answer

Expert verified
The factored form is \( (p+q-8)(p+q+8) \).

Step by step solution

01

- Identify the expression

Recognize the given expression: \( (p+q)^{2}-64 \). This is a binomial expression.
02

- Recognize the difference of squares

Notice that the given expression is in the form of \( a^2 - b^2 \), where \( a = (p+q) \) and \( b = 8 \), since \( 8^2 = 64 \).
03

- Apply the difference of squares formula

Utilize the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \). Substitute \( a = (p+q) \) and \( b = 8 \) into the formula. This gives us: \( ((p+q) - 8)((p+q) + 8) \).
04

- Simplify the factors

Simplify both binomials to finish the factorization process: \( (p+q-8)(p+q+8) \).

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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, recognizing a difference of squares is a crucial skill, especially when factoring polynomials. The difference of squares formula states that any expression of the form \(a^2 - b^2\) can be factored into \( (a - b)(a + b) \). For example, in the exercise \((p+q)^2 - 64\), we can see it fits this pattern because we have two perfect squares: \((p+q)^2\) and \(64\) (since \(8^2 = 64\)). By applying the difference of squares formula, we split the original expression into two binomial expressions, making it easier to solve.
Binomial Expression
A binomial expression is simply an algebraic expression that contains two terms. For example, \( p + q \) is a binomial because it has two terms, \(p\) and \(q\). When dealing with the difference of squares, each binomial factor resulting from the factorization process must be considered individually. For the given exercise, when we recognize \((p+q)^2 - 64\), this is a difference of two squares where our binomial terms are \((p+q)\). After applying the difference of squares formula, we break it down to two separate binomials: \((p+q - 8)\) and \((p+q + 8)\).
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler components, or 'factors,' that when multiplied together yield the original polynomial. The exercise provides a clear illustration of this process for the binomial expression \((p+q)^2 - 64\), which we identified as a difference of squares. By substituting \(a = (p+q)\) and \(b = 8\), and then applying the formula, we factorize the polynomial into \((p+q-8)(p+q+8)\). This breaks down the original polynomial into two simpler factors, making it easier to solve or further manipulate. Understanding these steps enhances one's ability to apply factorization in varied algebraic problems.

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