Problem 19 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ q^{2}-64 $$

Step-by-Step Solution

Answer

\(q^{2} - 64\) can be factored as \((q+8)(q-8)\)

Step 1: Identifying the expression

We have an expression here of the form \(a^2 - b^2\), which is a difference of squares. In this case, \(a=q\) and \(b=8\), as \(64\) is a perfect square and its square root is \(8\).

Step 2: Applying the difference of squares formula

The difference of squares formula is \(a^2 - b^2 = (a+b)(a-b)\). If we apply this formula to the given expression, we get: \[q^{2} - 64 = (q+8)(q-8).\]

Key Concepts

Difference of Squares

Factoring expressions sometimes involves recognizing special patterns like the difference of squares. This is a helpful technique in algebra that simplifies expressions by transforming them into a product of two binomials. The difference of squares formula is derived from the equation:

• \(a^2 - b^2 = (a+b)(a-b)\)

This formula states that a quadratic expression with two squared terms, separated by a subtraction sign, can be factored into the product of two binomial expressions. This method only works if both terms are perfect squares.
For example, in the expression \(q^2 - 64\), the terms \(q^2\) and \(64\) are perfect squares. Recognizing this pattern allows us to express the problem as a multiplication of \((q+8)(q-8)\), simplifying the algebraic process and making it easier to solve other equations that include this expression.

Polynomials

Polynomials are algebraic expressions composed of variables and coefficients, arranged in terms of powers and operations such as addition, subtraction, and sometimes multiplication. The general form for a polynomial is given by:

• \(a_n x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\)

where \(a_n\), \(a_{n-1}\), ..., and \(a_0\) are coefficients, and \(n\) represents non-negative integer exponents. The degree of a polynomial is the highest exponent of the variable in the expression.
The given expression \(q^2 - 64\) is a specific type of polynomial called a quadratic polynomial because the highest power of the variable \(q\) is 2. This polynomial is particularly important because it can be factored using methods such as the difference of squares, which help in solving polynomial equations more efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can represent numbers, variables, and operations. They form the foundation of algebra by providing a way to describe generalized numbers and operations abstractly. Each algebraic expression consists of:

• Variables: Symbols like \(x\), \(y\), \(z\), or in this case, \(q\), which represent unknown values.
• Constants: Number elements that stand for specific, unchanging values, such as 64 in the example \(q^2 - 64\).
• Operations: Include addition (+), subtraction (-), multiplication (*), and division (/).

Algebraic expressions can be manipulated through various rules of algebra, which include factoring, expanding, and simplifying.
Factoring is essential as it breaks down complex expressions into simpler parts, aiding in solving equations or simplifying calculations. In our example, knowing how to identify and factor a difference of squares allows us to rewrite the expression in a simpler, more workable form.