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Chapter 5: Problem 136

Factor each polynomial completely. $$3 a^{3}-24 a^{2}+48 a$$

Short Answer

Expert verified
\(3a(a - 4)^2\)

Step by step solution

01

Factor out the Greatest Common Factor (GCF)

Identify and factor out the greatest common factor from all the terms. For the polynomial \(3a^3 - 24a^2 + 48a\), the GCF is 3a. Therefore, factor 3a out of each term:\[3a(a^2 - 8a + 16)\].
02

Factor the Quadratic Expression

Now, factor the quadratic expression inside the parentheses, \(a^2 - 8a + 16\). Look for two numbers that multiply to 16 and add up to -8. These two numbers are -4 and -4. Thus, the quadratic expression can be written as:\[(a - 4)(a - 4) = (a - 4)^2\].
03

Write the Complete Factored Form

Combine the GCF factored out in Step 1 with the factored quadratic expression from Step 2. The completely factored form of the polynomial is:\[3a(a - 4)^2\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)
The first step in factoring polynomials is to identify the Greatest Common Factor (GCF). The GCF is the largest factor that is shared by all terms in the polynomial. For instance, in the polynomial given in the exercise, which is \(3a^3 - 24a^2 + 48a\), we need to find the GCF of the coefficients and the variable parts.
- The coefficients are 3, -24, and 48. The GCF of these numbers is 3. - For the variable part, each term includes the variable \(a\) (\(a^3, a^2, and a\)). The GCF of the variable part is \(a\).

Combining these, we get a GCF of \(3a\)


Thus, the given polynomial can be factored by extracting the GCF: \[ 3a(a^2 - 8a + 16) \]
Quadratic Expression
Once the GCF has been factored out, we are often left with a quadratic expression. A quadratic expression is typically in the form \(ax^2 + bx + c\). In the exercise, the quadratic expression we need to factor is \(a^2 - 8a + 16\).
The goal is now to factor this expression into two binomials. To do this, we need to find two numbers that multiply to the last constant term (16) and add up to the middle coefficient (-8).

Those two numbers are -4 and -4


Thus, we can write the quadratic expression as: \[(a - 4)(a - 4)\] Which can be simplified to: \[(a - 4)^2\]
Completely Factored Form
The final step is to combine all the factors obtained in the previous steps to write the polynomial in its completely factored form. We have already factored out the GCF \(3a\), and then factored the quadratic expression to get \((a - 4)^2\).

Now, putting it all together:

The completely factored form of the original polynomial \(3a^3 - 24a^2 + 48a\) is: \[3a(a - 4)^2\]
Each step is crucial in breaking down and simplifying the given polynomial into its fully factored state.

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