Problem 60 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$a^{3} b+10 a^{2} b^{2}+24 a b^{3}$$

Step-by-Step Solution

Answer

The completely factored form of the given expression \(\displaystyle a^3b + 10a^2b^2 + 24ab^3\) is \(\displaystyle ab(a + 4b)(1 + 6b)\).

Step 1: Identify the GCF

The first step is to determine the GCF for the given expression: \(a^{3}b + 10a^{2}b^{2} + 24ab^{3}\). We can see that the common factors for all three terms are \(a, b\). So, the GCF is \(ab\).

Step 2: Factor out the GCF

Next, we factor out the GCF, \(ab\), from the given expression: \[ ab(a^2 + 10ab + 24b^2) \] Now, we need to focus on factoring the expression inside the parenthesis: \(a^2 + 10ab + 24b^2\).

Step 3: Factor the Quadratic Expression

To factor the quadratic expression, we look for two binomials whose product is equal to the given quadratic expression. We can rewrite the expression as: \[ (a^2 + 4ab + 6ab + 24b^2) \] Now, we can use the distributive property to factor: \[ a(a + 4b) + 6b(a + 4b) \] Here, we notice that \((a + 4b)\) is a common factor in both terms. So, we can factor out this common factor:

Step 4: Write the Final Factorized Expression

After factoring out the common factor \((a + 4b)\), we get the final factorized form of the given expression: \[ ab(a + 4b)(1 + 6b) \] The given expression is now completely factored.

Key Concepts

Greatest common factor (GCF)

When working with polynomials, finding the greatest common factor (GCF) is a powerful first step in simplifying expressions. The GCF is the largest factor that divides each term of the polynomial without producing a remainder. It's like finding the biggest piece of a puzzle that fits each part of the picture.

To identify the GCF, examine each term in the polynomial independently. Consider both the numerical and variable parts:

• Look at all the numerical coefficients to find their greatest common divisor.
• For variable terms, select the smallest power of each variable present in every term.

For the expression \(a^3b + 10a^2b^2 + 24ab^3\):

• The numerical coefficients are 1, 10, and 24, whose GCF is 1 (since they have no common divisor other than 1).
• The variable terms have the factors of \(a\) and \(b\), with the smallest powers being \(a\) and \(b\), thus making the GCF \(ab\).

Once identified, the GCF is factored out of the expression, simplifying further operations.

Quadratic expression

A quadratic expression is one of the key structures in algebra, characterized mainly by terms with variables raised to the second power as its highest degree. These expressions are usually represented in the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

In the context of multi-variable expressions like \(a^2 + 10ab + 24b^2\), the expression still behaves quadratically relative to its components (considering \(ab\) as "x"). Here, the quadratic nature allows us to use techniques such as foil (First, Outside, Inside, Last) to rewrite the expression as:

• Finding two numbers that multiply to \(ac\) (where \(a\) and \(c\) are products found in the quadratic form) and add up to \(b\).
• These numbers guide the regrouping of the middle term and prepare the way for further simplification.

Such expressions can be seen in factorable forms wherein they resemble perfect square trinomials or can fit the form of a factorable quadratic equation, making them simpler to factor further.

Binomial factorization

Binomial factorization involves breaking down an expression into products of binomials, which are expressions with two terms each. This is a strategic step in simplifying quadratic or polynomial expressions and making them easier to solve or understand.

Consider the quadratic expression \((a^2 + 10ab + 24b^2)\) which is restructured into \((a + 4b) + 6b(a + 4b)\). Here, we can observe:

• Identifying a common factor in pairs or entire terms, often resulting from regrouping or expansion techniques.
• By factoring out \((a + 4b)\), the expression naturally decomposes into two binomial factors \((a + 4b)(1 + 6b)\).

Through careful manipulation of these binomial structures, we end up with a fully factorized expression \(ab(a + 4b)(1 + 6b)\). This broken down form makes the polynomial much easier to work with, especially in solving equations or performing operations.