Problem 46 - College Algebra Essentials

Question

$$\text { Factor the difference of two squares.}$$ $$x^{4}-1$$

Step-by-Step Solution

Answer

The factorized form of \(x^{4} - 1\) is \((x-1)(x+1)(x^{2}+1)\).

Step 1: Identify a and b

The given expression is \(x^{4}-1\). Referring back to the general form of difference of squares, \(a^{2}-b^{2}\), here \(a^{2}=x^{4}\) and \(b^{2}=1\). Therefore, \(a=x^{2}\) and \(b=1\). Thus, the initial difference of squares can be expressed as \((x^{2})^{2} - (1)^{2}\).

Step 2: Apply the formula

Next, apply the formula \(a^{2}-b^{2} = (a-b)(a+b)\). Substituting the values for \(a\) and \(b\), we get \((x^{2}-1)(x^{2}+1)\).

Step 3: Further Factorize if possible

The term \(x^{2}-1\) is still a difference of squares and can be factorized further. Here, for \(x^{2}-1\), \(a=x\) and \(b=1\). Apply the formula to get \((x-1)(x+1)\). The second term \(x^{2}+1\) cannot be factorized further as it is a sum of squares.

Step 4: Final Result

Thus, substituting the factorized form of \(x^{2}-1\) back into the equation, the factorization of the provided expression \(x^{4}-1\) is \((x-1)(x+1)(x^{2}+1)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that involves breaking down a polynomial into a product of simpler polynomials. Think of it like finding what numbers you can multiply together to get the original polynomial. This process is especially useful when solving equations, simplifying expressions, or finding zeros of a polynomial function.

For instance, a polynomial like x^2 - 9 can be factored into (x - 3)(x + 3). When factoring, we often look for patterns such as the difference of two squares, perfect square trinomials, or common factors among the terms.

The method of factoring will depend on the form of the polynomial and can include techniques like grouping, using the quadratic formula, synthetic division, or special formulas.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (such as addition, subtraction, multiplication, and division). These expressions can range from simple, with just a few terms, to very complex. A key element in working with algebraic expressions is understanding how to manipulate and simplify them.

For example, the expression 3x + 2y - 5x + y can be simplified by combining like terms to get -2x + 3y. It's important to recognize patterns and use algebraic properties to simplify and solve expressions. These properties include distributive, associative, and commutative properties.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as the product of its factors. These factors can be numbers, variables, or more complex expressions. Factorization can greatly simplify many problems in algebra, such as solving polynomial equations or integrating polynomials.

Factors are often found by looking for patterns within the polynomial or using specific methods like grouping, long division, or synthetic division. For example, the polynomial x^2 - 4 can be factorized to (x - 2)(x + 2) using the difference of two squares pattern.

Sum and Difference of Squares

The sum and difference of squares are special patterns in algebra that can be factorized using specific formulas. The difference of squares formula is a^2 - b^2 = (a - b)(a + b), which applies to any expression where two terms are squares and are subtracted.

On the other hand, the sum of squares, given by a^2 + b^2, generally cannot be factorized using real numbers. These formulas are particularly powerful because they apply universally and can simplify complex expressions into products of binomials.

As seen in the example x^4 - 1, recognizing it as (x^2)^2 - 1^2 allows us to use the difference of squares formula to factor it as (x^2 - 1)(x^2 + 1) and further factorize x^2 - 1 into (x - 1)(x + 1), giving us the final factorization of (x - 1)(x + 1)(x^2 + 1).