Problem 16 - Beginning and Intermediate Algebra

Question

Factor completely. $$ 16 b^{2}-24 b+9 $$

Step-by-Step Solution

Answer

The short answer is: \((4b - 3)^{2}\).

Step 1: Identify common factors

First, let's look at the coefficients: 16, -24 and 9. We can identify that there is no common factor between them, so we will move on to the next step.

Step 2: Check if it's a perfect square trinomial

Recall that a perfect square trinomial can be written as \((A+B)^{2}\) or \((A-B)^{2}\), where it takes the form \(A^2 \pm 2AB + B^2\). In our case, we have: \(16b^2 - 24b + 9\) We can rewrite this as: \((4b)^{2} - 2(4b)(3) + (3)^{2}\) Here, we can see that it matches the form of a perfect square trinomial with \(A = 4b\) and \(B = 3\), and it's a subtraction: \((4b)^2 - 2(4b)(3) + (3)^2 = (4b - 3)^2\) So, the complete factorization is:

Step 3: Final Answer

\((4b -3)^{2}\)

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of algebraic expression that can be factored into a binomial squared. This kind of trinomial takes the form

• \(A^2 + 2AB + B^2\), which factors to \((A + B)^2\)
• \(A^2 - 2AB + B^2\), which factors to \((A - B)^2\)

In other words, a perfect square trinomial is an expression consisting of a square of a term, plus or minus twice the product of two terms, plus the square of another term.
For example, in our exercise: the expression \(16b^2 - 24b + 9\) can be rewritten using the perfect square form. It fits the pattern \((4b)^2 - 2(4b)(3) + (3)^2\), where \(A = 4b\) and \(B = 3\).
Recognizing and working with perfect square trinomials can simplify the factoring process and is a useful tool in algebra.

Complete Factorization

Complete factorization involves breaking down an expression into the product of its simplest building blocks, largely into linear factors or irreducible quadratic factors. When factoring completely, it is crucial to check for perfect square trinomials, common factors, or even apply methods like trial and error, grouping, or special identities.
In the exercise, after determining there were no common factors such as greatest common divisors among the terms \(16\), \(-24\), and \(9\), we looked for patterns such as the perfect square trinomial. Once identified,

• it could be expressed as the square of a binomial: \((4b - 3)^2\)

This means the expression \(16b^2 - 24b + 9\) is completely factored when written as one binomial squared. Complete factorization ensures we get down to the simplest version of the expression, crucial for solving equations and finding solutions efficiently.

Quadratic Expressions

Quadratic expressions are algebraic expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(x\) represents an unknown variable. These expressions are particularly important because they appear frequently in various mathematical problems, such as

• polynomial equations,
• graphing parabolic curves, and
• modeling different types of motion.

Quadratics like \(16b^2 - 24b + 9\) showcase a specific form with terms including \(b^2\), \(b\), and a constant.
Understanding how to manipulate and factor these expressions is essential because it simplifies problem solving, allowing us to use techniques to find real-world solutions. By transforming quadratic expressions through methods like factoring, we can retrieve values that traditional solving methods may not reveal so easily.