Problem 126 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(x^{3}+5 x^{2}+4 x+20\)

Step-by-Step Solution

Answer

The expression is factored as \((x+5)(x^2+4)\).

Step 1: Group Terms

First, we split the polynomial into two groups to ease the factoring process: \( x^3 + 5x^2 \) and \( 4x + 20 \).

Step 2: Factor Each Group

Find the greatest common factor (GCF) of each group. For \( x^3 + 5x^2 \), the GCF is \( x^2 \), so it becomes \( x^2(x + 5) \). For \( 4x + 20 \), the GCF is \( 4 \), so it becomes \( 4(x + 5) \).

Step 3: Factor by Grouping

Now that each group contains \( (x + 5) \), factor out \( (x + 5) \) from the entire expression: \((x^2 + 4)(x + 5)\).

Step 4: Check for Further Factoring

Check each factor to see if it can be factored further. \( x^2 + 4 \) is a sum of squares and cannot be factored over the real numbers, so no further factoring is possible.

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is a crucial concept when it comes to factoring polynomials. It refers to the largest factor that is common to all terms in an expression. Finding the GCF is your first step in simplifying expressions by factoring.

To locate the GCF, check each term of the polynomial for common numerical coefficients and variable components. For instance, in the expression \(x^3 + 5x^2 + 4x + 20\), consider the separate terms: \(x^3 + 5x^2\) and \(4x + 20\). You'll notice that \(x^2\) is common in the first group \((x^3 + 5x^2)\), and the number 4 is common in the second group \((4x + 20)\). Thus:

• In the first group, the GCF is \(x^2\).
• In the second group, the GCF is 4.

Finding the GCF helps us break down expressions into more manageable parts, setting the stage for further factoring techniques like factoring by grouping.

Factoring by Grouping

Factoring by grouping is a technique used when you have a polynomial with four or more terms. The idea is to group terms together in pairs, then factor each pair separately using the GCF. Once each group is factored, if a common factor appears in each group, it can be factored out from the entire expression.

So, let’s look at our initial polynomial: \(x^3 + 5x^2 + 4x + 20\). We divide it into two groups: \(x^3 + 5x^2\) and \(4x + 20\). We then factor each group:

• For \(x^3 + 5x^2\), we factor out \(x^2\) to get \(x^2(x + 5)\).
• For \(4x + 20\), we factor out 4 to get \(4(x + 5)\).

Notice that \(x + 5\) is common in both groups. We can then factor \((x + 5)\) from the entire expression to get:\
\((x^2 + 4)(x + 5)\).

Factoring by grouping efficiently reduces complex expressions into simpler ones, making it easier to see additional factoring opportunities or confirm completeness.

Sum of Squares

While many polynomials can be factored further, certain expressions, like the sum of squares, remain in their simplest form over the real numbers. In our polynomial example, we've arrived at \(x^2 + 4\), which is a sum of squares.

A sum of squares differs from the more familiar difference of squares, which can be easily factored. Unlike the difference of squares—which can be expressed as \((a^2 - b^2) = (a - b)(a + b)\)—a sum of squares like \(x^2 + 4\) does not have a similar real number factorization. Its structure inherently resists any further factoring using real numbers.

This characteristic is due to the fact that for real numbers, there exist no two real numbers whose product is negative. Consequently, expressions like \(x^2 + 4\) do not factor further in real terms, and are considered unfactorable under real number constraints, marking the completion of the factoring process.