Problem 43 - A Graphical Approach to College Algebra

Question

Factor each difference of squares completely. $$25 s^{4}-9 t^{2}$$

Step-by-Step Solution

Answer

The expression factors to \((5s^2 - 3t)(5s^2 + 3t)\).

Step 1: Identify the Difference of Squares

The expression given is \(25s^4 - 9t^2\). Notice that it is in the form \(a^2 - b^2\), where \(a = 5s^2\) and \(b = 3t\). This is a difference of squares because both terms are perfect squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Applying this to our expression, we get:\[ (5s^2 - 3t)(5s^2 + 3t) \] This effectively factors the original expression into the product of two binomials.

Key Concepts

Factoring Techniques

When it comes to simplifying algebraic expressions, factoring can be a highly efficient method. Understanding how to break down expressions into products of simpler expressions can make solving equations much easier. In the case of the difference of squares, we leverage a special factoring technique.

The difference of squares formula, which is quite useful, states:

• \(a^2 - b^2 = (a - b)(a + b)\)

This formula works specifically for expressions where two terms are squared and separated by a minus sign. To apply this technique, you first identify the two square terms, as in the expression \(25s^4 - 9t^2\). Here, each term can be rewritten as a square: \((5s^2)^2\) and \((3t)^2\).

By recognizing the expression as a difference of squares, you are able to apply the formula and factor it into two binomial products: \((5s^2 - 3t)(5s^2 + 3t)\). This technique highlights the power of recognizing patterns in mathematical expressions.

Algebraic Expressions

Algebraic expressions form the foundation of algebra, encompassing both the operations and symbols used to convey mathematical ideas. These expressions can include numbers, variables, and operations like addition, subtraction, multiplication, and division.

Understanding the structure of an algebraic expression can significantly aid in its manipulation, such as in the exercise \(25s^4 - 9t^2\). Here, the variables \(s\) and \(t\) are used, and the expression features exponentiation, which indicates the power to which a number or variable is raised. Specifically, \(25s^4\) involves a variable \(s\) raised to the fourth power, while 9 must be accompanied by \(t^2\), emphasizing both are squared terms.

Working with algebraic expressions often requires us to factor, expand, and simplify, skills that are essential for solving equations effectively. Recognizing different types of expressions, such as the difference of squares, can reduce complexity, making problem-solving more manageable.

Binomials

In algebra, binomials are expressions consisting of two terms connected by either a plus or a minus sign. Binomials are fundamental in various algebraic operations, particularly factoring, which is applied in the exercise \((5s^2 - 3t)(5s^2 + 3t)\).

When factoring the expression \(25s^4 - 9t^2\), we use the difference of squares method to obtain two binomials. These binomials, \(5s^2 - 3t\) and \(5s^2 + 3t\), are products of terms that are independently simpler than the original expression.

• Each binomial consists of two distinct terms.
• They can be added to or multiplied with other expressions to form complex equations or inequalities.

Binomials are instrumental in solving quadratics, simplifying expressions, and polynomial expansion. Their simplicity is what makes them a key focus in algebra, allowing for straightforward operations and easier comprehension of more complex algebraic concepts.