Problem 49 - Algebra 1: Concepts and Skills

Question

COMMON FACTOR Factor the expression. $$ 16 w^{2}+80 w+100 $$

Step-by-Step Solution

Answer

The factored form of the expression \(16w^{2}+80w+100\) is \(4w(2w+5)^2.

Step 1: Identify the Greatest Common Factor

When factoring, first look for the greatest common factor (GCF) in all terms. The terms are \(16w^{2}\), \(80w\), and \(100\). The GCF of the numbers 16, 80, and 100 is 4. The variable \(w\) is present in the first two terms. Therefore, the GCF is \(4w\).

Step 2: Divide each term by the GCF

Next, each term is divided by the GCF, \(4w\). The expression then becomes: \(4w(4w^{2}+20w+25)\)

Step 3: Factoring quadratic expression

Now, the quadratic expression inside the parenthesis, \(4w^2+20w+25\), can be factored further to \((2w+5)(2w+5)\), since the numbers 2 and 5 add up to 20 (the coefficient of \(w\)) and multiply to give 25.

Step 4: Write the fully factored expression

Now, combining everything, the fully factored expression will be \(4w(2w+5)^2\).

Key Concepts

Greatest Common Factor

When tackling polynomial expression problems, finding the Greatest Common Factor (GCF) is often the first crucial step. The GCF is the largest factor that divides all the terms of the expression. Consider the polynomial given in the exercise:

• Terms: 16\(w^2\), 80\(w\), and 100.
• Numbers: 16, 80, and 100.
• Common variable: \(w\) for the first two terms.

To find the GCF of the numbers, consider their common divisors. Here, 4 is the greatest number that divides all three numbers evenly.
For variables, we choose the lowest power of \(w\), which is the single \(w\) (as it's present in the first two terms). Therefore, the GCF for this polynomial expression is \(4w\). Right from the start, identifying the GCF simplifies everything you do later!

Quadratic Expressions

Quadratic expressions are polynomial expressions that involve terms with variables raised to the power of two, typically written in the form \(ax^2 + bx + c\). These expressions crop up a lot in algebra, and learning to factor them is essential.
In our exercise, after extracting the GCF, we deal with the quadratic expression \(4w^2 + 20w + 25\). The key to factoring quadratics is looking for two numbers that both add to the coefficient of the first-degree term (in this case, 20) and multiply to the constant term (here, 25).

• Possible candidates: 2 and 5, as 2+5=20 and 5\(\times\)5=25.

This particular quadratic is a perfect square trinomial because it factors to \((2w+5)(2w+5)\) or \((2w+5)^2\). Recognizing these patterns makes factoring much easier.

Algebraic Expressions

Algebraic expressions encompass a wide array of mathematical constructs made up of variables, coefficients, and constants. Understanding how to manipulate these expressions is foundational in algebra.
Factoring, especially, involves rewriting a given algebraic expression as the product of its factors, which often simplifies complex problems. For instance:
Our exercise provided: \(16w^2 + 80w + 100\). By factoring out the GCF first, then exploiting the factors of the quadratic expression within, we reach a simpler and more usable form.

• Factoring process: First identify the GCF, leading us to \(4w\).
• Then, deal with the quadratic expression \(4w^2 + 20w + 25\), factoring it as \((2w+5)^2\).

Finally, we write it concisely as \(4w(2w+5)^2\). By breaking it down step by step, algebraic expressions become much less daunting, giving you a structured way to solve problems.