Problem 34 - 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. $$20 x^{3}+45 x$$

Step-by-Step Solution

Answer

The complete factorization is \(5x(4x^2 + 9)\).

Step 1: Identify Common Monomial Factor

The given polynomial is \(20x^3 + 45x\). Both terms contain the variable \(x\), so \(x\) is a common factor. Additionally, the coefficients 20 and 45 both have a greatest common divisor of 5.

Step 2: Extract the Common Monomial Factor

Factor out \(5x\) from each term. This gives: \[ 20x^3 + 45x = 5x(4x^2 + 9) \]

Step 3: Check for Further Factorization

Examine the polynomial \(4x^2 + 9\) to see if it can be factored further. Since this is a sum of squares and not a difference, it cannot be factored using real coefficients.

Step 4: Confirm the Complete Factorization

The polynomial \(5x(4x^2 + 9)\) is fully factored over the integers, as the expression \(4x^2 + 9\) cannot be further factored into integers.

Key Concepts

Monomial Factor

When working with polynomials, a crucial first step in the factorization process is to identify a monomial factor. A monomial factor is a single term that can be divided evenly into each term of the polynomial. In the provided polynomial, \(20x^3 + 45x\), notice that each term includes the variable \(x\). This means that \(x\) itself is a common factor.
Moreover, when dealing with the numerical coefficients like 20 and 45, you can find a greatest common divisor, which in this case is 5.
Combining both the numerical and variable parts gives you \(5x\) as the monomial factor to factor out from the polynomial.

Greatest Common Divisor

The greatest common divisor (GCD) plays a significant role in simplifying polynomials. For the coefficients 20 and 45, the GCD is the largest number that divides both without leaving a remainder. Using basic divisibility rules or a step-by-step approach, it's clear that 5 is that number.
Finding the GCD helps to streamline the factorization process by reducing the polynomial to its simplest form. Essentially, you start by dividing each term of the polynomial by the GCD to extract the common factor, a critical step before proceeding with further factorization attempts.

Sum of Squares

When the polynomial includes a term like \(4x^2 + 9\), it introduces a concept known as the sum of squares. Unlike the difference of squares, which can be factored easily into \((a-b)(a+b)\), the sum of squares \(a^2 + b^2\) does not have a straightforward factorization using real numbers.
In many real-world problems, attempts to factor a sum of squares may often lead students astray. In our case, \(4x^2 + 9\) cannot be factored further over the integers, maintaining its structure in the final factored expression.

Factor completely

Factoring completely means breaking down the polynomial into products of simpler polynomials that cannot be factored any further. Start by identifying and extracting any monomial factors, such as \(5x\) in our given exercise. This reduces the polynomial in stages.
After removing the monomial factor, what remains is \(4x^2 + 9\), which cannot be factored further, thus completing the factorization process. This ensures that the polynomial has been reduced to its simplest form possible over the integers.

• Always check for a common factor first.
• Attempt further factorization if possible, but recognize when you have reached a termination point, such as a sum of squares.