Problem 10 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$1-64 x^{2}$$

Step-by-Step Solution

Answer

\((1 - 8x)(1 + 8x)\

Step 1: Identify the difference of squares

First, look at the expression \(1 - 64x^{2}\) and recognize it as a difference of squares equation. The squares in this case are \(1\) (which is \(1^{2}\)) and \(64x^{2}\) (which is \((8x)^{2}\)).

Step 2: Apply the difference of squares formula

Now, apply the formula for the difference of squares, which states that \(a^2 - b^2 = (a - b)(a + b)\). Substituting \(1\) for \(a^2\) and \(64x^{2}\) for \(b^2\), we get \((1^{½} - (8x))(1^{½} + (8x))\).

Step 3: Write out the final factorization

Simplified, this becomes \((1 - 8x)(1 + 8x)\), which is the factorization of the original expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that involves breaking down polynomials into simpler components that, when multiplied together, yield the original polynomial. Think of it as a reverse multiplication process; it's analogous to finding out what ingredients went into a cake rather than baking one from scratch.

When working with the difference of two squares, a form of polynomial, the task is to identify terms that are perfect squares and to express them as the product of binomials. For instance, the expression \(1 - 64x^2\) factors into \(1 - 8x\) and \(1 + 8x\), since both 1 and \((8x)^2\) are square terms. Thus, understanding patterns and applying specific factoring formulas are pivotal components in mastering the factoring of polynomials.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They represent quantities in an abstract way, allowing for the description of relationships and the formulation of equations. In our example \(1 - 64x^2\), the expression is a condensed way of representing a quantity that varies with \(x\).

Recognizing the structure of algebraic expressions is key to understanding and solving polynomial equations. Each \(x\) carries a weight, affecting the value of the expression as \(x\) varies. Learning to manipulate these expressions through operations like factoring gives students powerful tools to solve a variety of problems in algebra.

Square Roots

The square root operation is fundamental in understanding the difference of two squares. It is the inverse operation of squaring a number. For instance, the square root of \(1\) is \(1\), and the square root of \(64x^2\) is \(8x\), because \(1^2 = 1\) and \(8x\) squared is \(64x^2\).

In the context of our example, recognizing that \(64x^2\) is a perfect square and that it can be rewritten as \(8x\) squared allows us to apply the formula for the difference of two squares effectively. Understanding how square roots relate to their squares can simplify the process of factoring considerably.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. Factors are polynomials of a lower degree that, when multiplied together, give the original polynomial. The difference of two squares is a special form of factorization where we express a binomial \(a^2 - b^2\) as the product \(a - b\) and \(a + b\).

By identifying and applying specific factorization patterns, like the difference of two squares, you can simplify complex expressions and solve algebraic equations more easily. In the exercise \(1 - 64x^2\), we used this method to factor the polynomial, resulting in \(1 - 8x\) and \(1 + 8x\), making it a clear demonstration of polynomial factorization.