Problem 48 - A Graphical Approach to College Algebra

Question

Factor each difference of squares completely. $$m^{4}-81$$

Step-by-Step Solution

Answer

The complete factorization is \((m^2 + 9)(m + 3)(m - 3)\).

Step 1: Identify Difference of Squares

Recognize that the expression given is in the form of a difference of squares: \(a^2 - b^2 = (a + b)(a - b)\).In our expression, let us express \(m^4 - 81\) as \((m^2)^2 - 9^2\), where \(a = m^2\) and \(b = 9\).

Step 2: Apply the Difference of Squares Formula

Using the identified values of \(a = m^2\) and \(b = 9\), substitute them into the difference of squares formula:\[(m^2)^2 - 9^2 = (m^2 + 9)(m^2 - 9)\].

Step 3: Factor Further if Possible

Observe that the term \(m^2 - 9\) is itself a difference of squares and can be factored further:\(m^2 - 9 = (m)^2 - 3^2 = (m + 3)(m - 3)\).Thus, we rewrite the original factorization as:\((m^2 + 9)(m + 3)(m - 3)\).

Step 4: Final Verification

Ensure that all expressions are completely factored. In this case, \(m^2 + 9\) cannot be factored further using real numbers as it is not a difference of squares.The complete factorization of \(m^4 - 81\) is: \((m^2 + 9)(m + 3)(m - 3)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into a product of simpler factors. This process is significant in algebra and is often guided by specific formulas or techniques. One common method is factoring through the difference of squares. This method is applied when an expression is in the form of \( a^2 - b^2 \), which can be expressed as \((a + b)(a - b)\).
In the given exercise, the expression \( m^4 - 81 \) is a perfect example of this. By rewriting it as \((m^2)^2 - 9^2\), you can identify it in the form of \( a^2 - b^2 \), where \( a = m^2 \) and \( b = 9 \). Using the difference of squares formula, one can express it as a product of two binomials: \((m^2 + 9)(m^2 - 9)\).
Factoring polynomials not only simplifies the expressions but also makes solving equations easier as it reduces the complexity of the problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like add, subtract, multiply, and divide). Understanding these expressions is fundamental in algebra as they form the basis for more advanced computations.
In the expression \( m^4 - 81 \), each component plays a crucial role. The term \( m^4 \) indicates a variable raised to the fourth power, and \( 81 \) is a constant represented as \( 9^2 \). Concepts like factoring use strategies such as the difference of squares to simplify these expressions by splitting them into products of simpler factors. This understanding is essential when examining expressions in algebraic contexts, as it allows one to handle calculations more efficiently.
By recognizing how the expression \( m^4 - 81 \) can be broken down, students begin to appreciate how algebraic structures can be manipulated and used to solve equations and real-world problems effectively.

College Algebra

College algebra covers a range of topics that equip students with essential skills for tackling more complex mathematical scenarios. One of the essential skills learned in this course is factoring expressions like differences of squares, which appear frequently in polynomial equations.
Students are taught to see the expression \( m^4 - 81 \) not just as numbers and variables but as a pattern that can be simplified. By breaking this pattern down through factoring into \((m^2 + 9)(m + 3)(m - 3)\), they learn to manage equations by making them more straightforward to solve.
This skill is vital in college algebra because it lays the groundwork for understanding more advanced courses, where these fundamentals are built upon. The ability to factor efficiently aids in solving equations and inequalities, analyzing functions, and understanding the properties and behaviors of various algebraic structures. Hence, mastering these foundational techniques is crucial for any student progressing through a modern mathematics curriculum.