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Chapter 5: Problem 84

Factor each polynomial completely. $$ a^{3}-6 a^{2}+9 a $$

Short Answer

Expert verified
The polynomial factors to \[a(a - 3)^{2}\].

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Look at each term of the polynomial and find the greatest common factor (GCF) that can be factored out. In the polynomial \(a^{3} - 6a^{2} + 9a\), the GCF is \(a\).
02

Factor out the GCF

Divide each term in the polynomial by the GCF and factor it out: \[a \times (a^{2} - 6a + 9)\]
03

Factor the quadratic expression

Now, focus on the quadratic polynomial \(a^{2} - 6a + 9\) inside the parentheses. Look for factors of 9 that add up to \(-6\). The factors are \(-3\) and \(-3\). Hence, \(a^{2} - 6a + 9 = (a - 3)(a - 3) = (a - 3)^{2}\)
04

Write the final factored form

Combine the factored terms to get the final factored form of the polynomial: \[a(a - 3)^{2}\]

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) is essential when factoring polynomials. It involves identifying the largest factor that all terms in the polynomial share. In the polynomial \(a^{3} - 6a^{2} + 9a\), we notice that each term contains the variable \(a\). Thus, the GCF is \(a\).
By factoring out the GCF, we simplify the polynomial into a more manageable form, making further factoring easier. Factoring out \(a\) from each term, we get:
\[a \times (a^{2} - 6a + 9) \]
Factoring the GCF first is crucial because it reduces the complexity of the polynomial, allowing us to focus on the next steps.
Quadratic Expression
A quadratic expression is a polynomial of degree 2, typically in the form \(ax^{2} + bx + c\). In our problem, after factoring out the GCF, we are left with the quadratic polynomial \(a^{2} - 6a + 9\).
This expression can be factored further by finding two numbers that multiply to the constant term (9) and add up to the coefficient of the middle term (-6). Here, those numbers are -3 and -3, so we can rewrite the quadratic expression as:
\[(a - 3)(a - 3)\]
Thus, \(a^{2} - 6a + 9\) factors into \((a - 3)^{2}\). Understanding how to handle quadratic expressions is vital, as they frequently appear in various polynomial factoring problems.
Factored Form
The factored form of a polynomial shows it broken down into products of simpler polynomials or constants. For our polynomial \(a^{3} - 6a^{2} + 9a\), after completing all the factoring steps, we get:
\[a(a - 3)^{2} \]
In this factored form, it is easier to identify the roots or solutions of the polynomial. The roots are the values of \(a\) that make the polynomial equal to zero. Here, setting \(a(a - 3)^{2} = 0\) gives us the roots \(a = 0\) and \(a = 3\).
Factored forms are not only simpler to work with but also provide insight into the polynomial's properties, such as its roots and behavior at those points.

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