Problem 27 - Beginning and Intermediate Algebra

Question

Factor completely. $$2 p^{4}-24 p^{3}+72 p^{2}$$

Step-by-Step Solution

Answer

The completely factored expression of \(2p^4 - 24p^3 + 72p^2\) is \(2p^2(p-6)^2\).

Step 1: Identify the GCF

First, we need to find the Greatest Common Factor (GCF) for all terms in the expression. The GCF of \(2p^4, -24p^3, 72p^2\) is \(2p^2\).

Step 2: Factor out the GCF

Now, we factor out the GCF (\(2p^2\)) from all terms in the expression. \(2p^2(p^2 - 12p + 36)\)

Step 3: Factor the quadratic expression

Next, we need to factor the quadratic expression inside the parentheses. We are looking for two numbers that multiply to give \(36\) and add up to \(-12\). These numbers are \(-6\) and \(-6\). Thus, the quadratic expression can be factored into \((p-6)(p-6)\) or \((p-6)^2\).

Step 4: Write the final factored expression

Finally, we can write the fully factored expression by incorporating the result from Step 3 into the expression from Step 2. The factored expression is: \(2p^2(p-6)^2\) So, the completely factored expression of the given exercise is \(2p^2(p-6)^2\).

Key Concepts

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is crucial when factoring polynomials. The GCF of a set of terms is the largest polynomial that divides each term without leaving a remainder. Identifying the GCF simplifies the process of factoring because it reduces the polynomial to a simpler form. In our example problem with the polynomial \(2p^4 - 24p^3 + 72p^2\), we begin by examining each term separately:

• \(2p^4\)
• \(-24p^3\)
• \(72p^2\)

We look for the highest power of \(p\) that is common to all terms, which is \(p^2\), and the greatest numerical factor common to the coefficients, which is 2. This means the GCF for this polynomial is \(2p^2\). By factoring out \(2p^2\) from the expression, we significantly simplify the polynomial, setting the stage for further factoring.

Quadratic Expression

A quadratic expression is a polynomial of degree 2, which generally takes the form \(ax^2 + bx + c\). In the exercise at hand, after factoring out the GCF, we're left with a quadratic expression \(p^2 - 12p + 36\). The challenge here is to factor this expression into two binomials.

We apply the method of finding two numbers that multiply to the constant term (in this case 36) and add to the linear coefficient (in this case -12). After examining potential pairs, we discover that both numbers are -6. Hence, \(-6 \, \text{and} \, -6\) satisfy both conditions:

• -6 \(-6 = 36\)
• -6 + -6 = -12

So, the quadratic expression \(p^2 - 12p + 36\) can be factored as \((p-6)^2\), revealing the roots of the quadratic as both 6.

Factored Expression

A factored expression is the result of factoring a polynomial completely, expressing it as a product of simpler terms. The goal of factoring is to break down complex expressions into their building blocks, or factors.

For our original polynomial expression \(2p^4 - 24p^3 + 72p^2\), once we've factored out the GCF \(2p^2\) and further factored the quadratic expression to \((p-6)^2\), our fully factored expression comes out to:

• \(2p^2(p-6)^2\)

This form is the simplest expression possible, making it easier to analyze and understand the properties and roots of the polynomial. Each part of the expression represents a specific factor: \(2p^2\) is the GCF, and \((p-6)^2\) is the factored quadratic, showing that the factor \(p-6\) repeats, indicating a double root at \(p = 6\).