Problem 29 - Algebra for College Students

Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$16 x^{2}+25$$

Step-by-Step Solution

Answer

The polynomial is not factorable over the integers.

Step 1: Identify the Type of Polynomial

The given polynomial is \( 16x^2 + 25 \). It has two terms and is a sum of squares: \( (4x)^2 + 5^2 \).

Step 2: Check for a Common Monomial Factor

Inspect the terms for a common factor. Here, 16 and 25 do not share any common factors other than 1, nor does the polynomial have a factor of \( x \) common to both terms.

Step 3: Consider Special Factoring Formulas

The polynomial is a sum of squares, which does not have a factoring formula over the integers. Sum of squares cannot be factored like difference of squares \((a^2 - b^2 = (a+b)(a-b))\).

Step 4: Conclude if Polynomial is Factorable

Since there is no integer factorization for the sum of squares in this form, \( 16x^2 + 25 \) cannot be factored over the integers. It is completely factored as given.

Key Concepts

Sum of Squares

In mathematics, the sum of squares refers to the addition of two squared terms. For instance, the polynomial \(16x^2 + 25\) can be recognized as a sum of squares because it is composed of the squares of \((4x)^2\) and \(5^2\). When you have a sum of squares like this, it's essential to know that no direct factoring formula exists over the integers — unlike the difference of squares, which can be expressed as \((a^2 - b^2 = (a+b)(a-b))\).

• Recognition: Sum of squares are expressed as \(a^2 + b^2\), where neither \(a\) nor \(b\) equals zero.
• Factoring: Sum of squares cannot be broken down into smaller integer factors.
• Characteristics: Recognizing a sum of squares helps avoid unnecessary factoring attempts.

Understanding sum of squares can help determine whether a polynomial can be factored using integers. This knowledge simplifies the process of tackling polynomial problems by indicating that factors do not exist within integer boundaries.

Common Monomial Factor

When factoring polynomials, starting with identifying any common monomial factor is a good strategy. A monomial factor is a single term that can be extracted from each term of the polynomial.
In the polynomial \(16x^2 + 25\), we check both terms, \(16x^2\) and \(25\), to see if there's a number or variable that can divide them evenly. The greatest common monomial factor here is \(1\), since neither term shares a variable \(x\) or any integers other than 1.

• Search for: Check if terms can be divided evenly by the same integer or variable.
• Importance: Reducing each term by a common factor simplifies the polynomial.
• Example: In \(4x^2 + 8x = 4x(x+2)\), \(4x\) is the common monomial factor.

Even when no common monomial factor is present, reaching this conclusion confirms either complete factorization or indicates more applicable factoring methods should be used.

Integer Factorization

Integer factorization involves expressing a number or an expression as a product of its integer factors. In polynomial factorization, this often involves breaking down terms into basic components that are integers.
For the polynomial \(16x^2 + 25\), integer factorization is not possible due to its form as a sum of squares. Typically, integer factorization would involve using techniques like factoring a common monomial or identifying special factoring patterns. However, the sum of squares remains unfactorable into integers.

• Definition: Expressing an expression as the product of integers.
• Challenges: Not all expressions can be factorized into integer factors.
• Limitation: Sum of squares like \(16x^2 + 25\) cannot use integer factorization.

The conclusion with expressions like \(16x^2 + 25\) is crucial to save time, revealing that without integer solutions, the expression stands as given, or might require complex or root-based solutions for further factorization.