Problem 69 - College Algebra with Modeling and Visualization

Question

Factor the expression completely, if possible. \(a^{2}+4 b^{2}\)

Step-by-Step Solution

Answer

The expression \(a^2 + 4b^2\) cannot be factored further over the reals.

Step 1: Identify the Expression Type

The given expression is \(a^2 + 4b^2\). This is a sum of two perfect squares.

Step 2: Recall Sum of Squares Factoring

Sum of squares does not have a straightforward factorization like the difference of squares. There is no general formula to factor \(a^2 + b^2\) over the real numbers.

Step 3: Check for Common Factors

The expression \(a^2 + 4b^2\) has no common factor other than 1. Thus, it cannot be factored further by finding a greatest common factor.

Step 4: Consider Complex Factorization

If considering complex numbers, the expression \(a^2 + 4b^2\) can be rewritten using complex numbers as \((a + 2bi)(a - 2bi)\) where \(i\) is the imaginary unit. However, the problem asks for real factorization.

Step 5: Final Conclusion

Since no real factorization is possible and the question asks for real-factorization, the expression \(a^2 + 4b^2\) is already in its simplest form over the reals.

Key Concepts

Sum of Squares

In algebra, a sum of squares is an expression of the form \(a^2 + b^2\). Unlike the difference of squares, which can be factored easily as \((a + b)(a - b)\), sum of squares cannot be factored using real numbers alone in the same straightforward manner. This is because there is no factorization of \(a^2 + b^2\) that results in factors composed solely of real numbers.
This concept often confounds students because it's common to find factorable counterparts such as \(a^2 - b^2\). It's important to grasp that sometimes expressions like \(a^2 + 4b^2\) are already in their simplest form when considering only real numbers.
Some expressions require stepping outside of real numbers for factorization, but the sum of squares remains unchanged if confined to the reals.

Complex Factorization

Complex factorization involves using complex numbers to express a sum of squares in a product of two factors. In the case of the sum of squares, although factorization over real numbers isn’t feasible, factorization can occur when using complex numbers.
Consider the expression \(a^2 + 4b^2\). This can be rewritten using complex numbers as

• \( (a + 2bi)(a - 2bi) \)

where \(i\) is the imaginary unit, defined by the property \(i^2 = -1\). Complex factorization transforms sum of squares expressions into products involving complex conjugates.
Understanding this method is useful in advanced mathematics, as it opens the door to a broader set of solutions, especially in fields like engineering and physics where complex numbers are frequently used.

Real Numbers in Algebra

Real numbers are fundamental in algebra. They include all the numbers you typically count with, including rational numbers like 3 and -1/2, and irrational numbers like \(\sqrt{2}\) and \(\pi\). In algebra, we often aim to express solutions or transformations using real numbers.
However, while dealing with expressions like \(a^2 + 4b^2\), real numbers set a limitation. Since such expressions can't be split into simpler terms over real numbers (as is possible with difference of squares), they highlight the importance and boundaries of real numbers in factorization.
When faced with algebraic expressions, understanding the realm of real numbers helps in defining what isn’t possible, such as recognizing when an expression is already simplified if it’s a sum of squares.