Problem 78 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$4 x^{2}+16$$

Step-by-Step Solution

Answer

The polynomial factors to \(4(x^2 + 4)\); it is not factorable further over the integers.

Step 1: Identify the Polynomial and Recognize the Common Factor

The given polynomial is \(4x^2 + 16\). First, we look for a common factor among the terms. Both terms, \(4x^2\) and \(16\), have a common factor of 4.

Step 2: Factor out the Common Factor

We factor out 4 from each term: \(4(x^2 + 4)\). Now the polynomial is simplified into 4 multiplied by another expression.

Step 3: Analyze the Quadratic Expression Inside the Parentheses

Check the expression \(x^2 + 4\). Notice that it doesn't factor further over the integers since it is the sum of a square and a number, not a difference of squares or a perfect square trinomial.

Step 4: Conclusion on Complete Factoring

Since \(x^2 + 4\) does not factor further using integer values, the factorization of the polynomial \(4x^2 + 16\) is complete. It is \(4(x^2 + 4)\).

Key Concepts

Common Factor

When factoring polynomials, the first key concept to grasp is the idea of a "common factor." A common factor is a number or variable that divides each term of a polynomial without leaving a remainder. Identifying a common factor can simplify a polynomial and make it easier to factor completely. In the original exercise, the polynomial given was \(4x^2 + 16\). Both terms, \(4x^2\) and \(16\), share the number 4 as a common factor. This is because the number 4 divides evenly into both 4 (from \(4x^2\)) and 16.

To factor out the common factor:

• Identify the greatest number that divides all coefficients and variables of the polynomial terms.
• Divide each term by this common factor.
• Write the polynomial as a product of the common factor and the resulting simplified expression.

In our example, factoring out the number 4 from \(4x^2 + 16\) results in \(4(x^2 + 4)\). This step often reveals simpler patterns or expressions that can further aid in complete factorization.

Quadratic Expression

Understanding quadratic expressions is crucial in the world of polynomials. A quadratic expression is any polynomial of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. In our exercise, the expression \(x^2 + 4\) falls under this category as it is in the general form of a quadratic. However, there’s a twist. The expression \(x^2 + 4\) lacks a linear term (\(bx\) part), making it a sum of squares.

Typically, factoring a quadratic expression involves finding two binomials that multiply to give you the original quadratic. However:

• The sum of squares like \(x^2 + 4\) does not factor over the integers since it's neither a perfect square trinomial nor a difference of squares.
• For this particular expression, further integer factorization isn't possible, which is why the factorization process stops here.

Recognizing these unique characteristics of quadratic expressions helps in determining if further factorization is possible.

Integer Factorization

When dealing with polynomials, integer factorization is a method of expressing a polynomial as a product of integers and smaller polynomials. The essential goal is to break down complex expressions into simpler, integer-factored components. This practice often involves identifying patterns, such as common factors or particular polynomial identities.

In the problem at hand, the polynomial \(4x^2 + 16\) was factored partly by factoring out the greatest common integer, resulting in \(4(x^2 + 4)\). The next step typically would involve attempting to factor the expression inside the parentheses over the integers. However:

• The expression \(x^2 + 4\) cannot be factored with integer values due to its nature as a sum of squares.
• Whenever you encounter expressions that can't be further broken down into integer components, you conclude the factorization process.

This highlights the importance of understanding integer factorization as it helps to determine when a polynomial has reached its simplest factored form with integer values.