Problem 63 - Beginning and Intermediate Algebra

Question

Factor completely. $$r^{4}-1$$

Step-by-Step Solution

Answer

The completely factored expression is \((r - 1)(r + 1)(r^2 + 1)\).

Step 1: Identify the difference of squares

The given expression can be viewed as a difference of squares, with \(r^4\) as the square of \(r^2\) and 1 as the square of 1: \[r^4 - 1 = (r^2)^2 - (1)^2\]

Step 2: Factor as a difference of squares

We will now factor the expression as a difference of squares: \[(r^2)^2 - (1)^2 = (r^2 - 1)(r^2 + 1)\]

Step 3: Factor the first term further

The expression \(r^2 - 1\) is also a difference of squares, as \(r^2\) is the square of r and 1 is the square of 1: \[r^2 - 1 = (r)^2 - (1)^2\]

Step 4: Factor the expression completely

We will now factor the expression \(r^2 - 1\) as a difference of squares and combine it with the expression \(r^2 + 1\) from step 2: \[(r)^2 - (1)^2 = (r - 1)(r + 1)\] \[(r^4 - 1) = (r^2 - 1)(r^2 + 1) = (r - 1)(r + 1)(r^2 + 1)\] The completely factored expression is \((r - 1)(r + 1)(r^2 + 1)\).

Key Concepts

Difference of Squares

The term "difference of squares" refers to an algebraic expression in the form of \(a^2 - b^2\). This can be easily factored into \((a - b)(a + b)\). Understanding why this method works is crucial. If you expand \((a - b)(a + b)\), you get

• \(a^2\) from \(a \times a\)
• \(-ab + ab\) which cancel each other out
• \(-b^2\) from \(-b \times b\)

Thus, you're left with \(a^2 - b^2\). In our specific problem, \(r^4 - 1\), we can set \(a = r^2\) and \(b = 1\), fitting perfectly into our formula. This method simplifies expressions and often reduces them into a product of simpler polynomials.

Factoring Quadratics

Factoring quadratics is essential in breaking down complex expressions into simpler ones. A quadratic equation is one that can be expressed in the form \(ax^2 + bx + c = 0\). When we factor quadratics, we search for two numbers whose product equals \(c\) and whose sum equals \(b\). In the expression \(r^2 - 1\), it's a special quadratic that belongs to the differences of squares category too, so it factors as \((r - 1)(r + 1)\).
Quadric equations might not always look like quadratics at first. Like in this exercise, \(r^4 - 1\), factors into two quadratics, \(r^2 - 1\) and \(r^2 + 1\). This sequence of identifying quadratics within polynomials is a key skill in factoring.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators like addition or multiplication. Expressions such as polynomials are central in algebra. Factoring them helps solve equations or simplify fractions.
When factoring \(r^4 - 1\), understanding the structure and components within an algebraic expression is vital. Recognizing parts of an expression that can be rewritten using known factoring techniques, such as the difference of squares, leads to a clearer path to a solution.
Getting comfortable with algebraic expressions requires practice in spotting opportunities to simplify them. In our problem, starting with the expression \((r^4 - 1)\), we identified nested differences of squares that allowed us to factor step by step, ultimately simplifying it to \((r - 1)(r + 1)(r^2 + 1)\). Mastery of this skill boosts confidence in tackling more complex algebraic problems.