Problem 135 - Introductory Algebra for College Students

Question

The polynomial \(4 x^{2}+100\) is the sum of two squares and therefore cannot be factored. If the general factoring strategy is used to factor a polynomial, at least two factorizations are necessary before the given polynomial is factored completely.

Step-by-Step Solution

Answer

The polynomial \(4x^2 + 100\) is correctly identified as a sum of squares, in the form of \((2x)^2 + (10)^2\). Based on sum of squares properties, it cannot be factored with real numbers.

Step 1: Recognize the Identity

Identify the type of expression. The given polynomial \(4x^2 + 100\) can be observed to be akin to the sum of squares algebraic identity \(a^2 + b^2\).

Step 2: Express as Sum of Squares

Rewrite the polynomial \(4x^2 + 100\) in the form \(a^2 + b^2\) where a and b are the terms that result in the given polynomial. We can rewrite the equation as \((2x)^2 + (10)^2\).

Step 3: Evaluate Factoring Possibility

A sum of squares (\(a^2 + b^2\)) cannot be factored further using real numbers, as this format does not allow for the use of the difference-of-squares formula \(a^2 - b^2 = (a-b)(a+b)\). Therefore, it is concluded that the given polynomial cannot be factored.

Key Concepts

Understanding the Sum of Squares

The term 'sum of squares' refers to an algebraic expression that consists of two perfect squares added together. In the context of polynomials, it usually appears in the form of a^2 + b^2, where a and b represent any algebraic terms.

In our example, we're looking at the polynomial 4x^2 + 100, which is a classic instance of a sum of squares. It represents the addition of two squared terms: (2x)^2 and 10^2. From a geometric perspective, you can think of these terms as areas of two squares, one with side 2x and the other with side 10.

It's crucial to recognize that a sum of squares cannot be factored over the set of real numbers, which is why our given polynomial 4x^2 + 100 does not factor further. This property is what distinguishes it from other forms like the difference of squares, which is factorable.

Applying Algebraic Identities

Algebraic identities are equations that are universally true for all values of the variables involved. These are the backbone of simplifying algebraic expressions and solving equations. For example, the identity (a + b)^2 = a^2 + 2ab + b^2 is an algebraic identity that represents the squared sum of two terms.

In the context of our exercise, although the sum of squares 4x^2 + 100 resembles an algebraic identity, it does not conform to factorable identities like (a + b)^2 due to the absence of the middle term 2ab. Recognizing correct identities and understanding how they are structured can save much time and effort when trying to factor polynomials, and in the case of our example, it helps understand why no further factoring can be done.

The Difference of Squares

In contrast to the sum of squares, the 'difference of squares' is a widely used algebraic identity in factoring polynomials. It takes the form a^2 - b^2 and can be factored into (a - b)(a + b). This identity is applicable because it turns a binomial expression into the product of two binomials.

For instance, if our initial polynomial had been 4x^2 - 100, this would align with the difference of squares identity, and we could factor it as (2x - 10)(2x + 10). This factoring is possible because subtraction allows for the creation of factors that, when multiplied, return the original expression.

Understanding the difference between a sum of squares, which is non-factorable, and a difference of squares, which is factorable, is fundamental in algebra. This concept often becomes a critical stepping-stone for students as they learn to factor more complex polynomials.