Problem 35 - Elementary and Intermediate Algebra

Question

Factor. \((r-s)^{2}-t^{4}\)

Step-by-Step Solution

Answer

The factorization of \((r-s)^2 - t^4\) is \((r-s - t^2)(r-s + t^2)\).

Step 1: Identify the Difference of Squares

The expression \((r-s)^2 - t^4\) is a difference of squares. Recall that the difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = (r-s)\) and \(b = t^2\), returning the equation \((r-s)^2 - (t^2)^2\).

Step 2: Apply the Difference of Squares Formula

Using the identity for the difference of squares, rewrite the expression \((r-s)^2 - t^4\) as two binomials: \((r-s - t^2)(r-s + t^2)\).

Step 3: Verify the Factorization

Expand \((r-s - t^2)(r-s + t^2)\) to verify the solution: \((r-s)^2 + (r-s)t^2 - (r-s)t^2 - t^4\). Notice the middle terms cancel, simplifying to \((r-s)^2 - t^4\). This confirms the factorization is correct.

Key Concepts

Difference of Squares

The difference of squares is an essential concept in algebra, mainly because it offers a straightforward way to factor specific expressions. The difference of squares refers to an expression in the form of \(a^2 - b^2\). It can be rewritten as a product of two binomials: \((a-b)(a+b)\). This pattern occurs because the middle terms cancel when multiplied out. Thus, this method provides an efficient way to simplify expressions involving squares.
When applying the difference of squares, first identify both parts of the expression as perfect squares. For instance, in \((r-s)^2 - t^4\), observe that both \((r-s)^2\) and \(t^4\) are squares. Replace them in the formula where \(a = (r-s)\) and \(b = t^2\). This understanding helps in transforming a complex expression into simpler factors, which is invaluable, especially in solving equations.

Binomials

Binomials are algebraic expressions that consist of two terms separated by either a plus or minus sign. Examples include \(x + y\) and \(x - y\). They are fundamental in various areas of algebra, including polynomial operations and factoring.
In the context of factoring the expression \((r-s)^2 - t^4\), using the difference of squares formula, we form the binomials \((r-s - t^2)\) and \((r-s + t^2)\). Binomials become critical as they can often be simplified or expanded depending on the need of the algebraic operation. Furthermore, understanding binomials is crucial for solving equations, simplifying expressions, and performing various algebraic manipulations.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically expressed in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In the specific exercise at hand, even though it might not look like a classic quadratic form, the idea of evaluating and simplifying expressions using the difference of squares plays into quadratic principles.
Though \((r-s)^2 - t^4\) itself isn't a quadratic expression in the traditional sense, the structure of such expressions often involves techniques similar to those used in factoring quadratics. Understanding quadratic expressions provides foundational knowledge needed for more complex algebraic concepts, including finding roots, solving equations, and factoring expressions. As you work with quadratic expressions, you become more familiar with patterns and methods that are applicable in broader mathematical contexts.