Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-8,\) write each polynomial as the product of its greatest common monomial factor and a polynomial. $$ 6 a^{4}-3 a^{3}+9 a^{2} $$

Step-by-Step Solution

Answer

The polynomial is factored as \(3a^2(2a^2 - a + 3)\).

Step 1: Identify the coefficients and terms

Examine the polynomial \(6a^4 - 3a^3 + 9a^2\). The coefficients are \(6\), \(3\), and \(9\). The terms are \(6a^4\), \(-3a^3\), and \(9a^2\).

Step 2: Determine the greatest common factor (GCF) of the coefficients

Find the greatest common factor of \(6\), \(3\), and \(9\). The factors are: \(6 = 2 \times 3\), \(3 = 3\), \(9 = 3 \times 3\). The common factor is \(3\).

Step 3: Determine the lowest power of the variable

Each term has a power of \(a\): \(a^4\), \(a^3\), and \(a^2\). The lowest power among them is \(a^2\).

Step 4: Combine the GCF of coefficients and variable

The greatest common monomial factor is \(3a^2\) because it combines the GCF of the coefficients \(3\) and the lowest power of the variable \(a^2\).

Step 5: Factor out the GCF from the polynomial

Divide each term in the polynomial \(6a^4 - 3a^3 + 9a^2\) by the GCF \(3a^2\): \(\frac{6a^4}{3a^2} = 2a^2\), \(\frac{-3a^3}{3a^2} = -a\), \(\frac{9a^2}{3a^2} = 3\).

Step 6: Write as a product of GCF and another polynomial

Express the original polynomial as \(3a^2 (2a^2 - a + 3)\).

Key Concepts

Understanding the Greatest Common Monomial Factor

When dealing with polynomials, one essential concept is recognizing the greatest common monomial factor (GCMF), which uses both the numerical coefficients and the variable parts of terms.
To find the GCMF, you first look at all terms in a polynomial and identify their coefficients and powers of variables separately.
For our example polynomial, \(6a^4 - 3a^3 + 9a^2\), we notice that each coefficient has a common divisor.
The coefficients \(6\), \(3\), and \(9\) all share \(3\) as a common factor. Next, examine the powers of \(a\): here, the smallest power that appears is \(a^2\).

• GCMF of coefficients: \(3\)
• Lowest variable power across terms: \(a^2\)

Combine these to get the greatest common monomial factor, which in this case is \(3a^2\). This is essentially the largest term that divides each term of the polynomial without leaving a remainder.

Finding the Greatest Common Factor

The greatest common factor (GCF) is crucial in breaking down complex expressions into simpler components. It refers specifically to the highest number that divides all the coefficients in the polynomial.
For our example, the coefficients were 6, 3, and 9.
Let's break these down:

• \(6 = 2 \times 3\)
• \(3 = 3\)
• \(9 = 3 \times 3\)

The common factor for these numbers is \(3\), which defines the numeric part of our GCMF. Once you've established this, remember to pair it with the variable's lowest power found in the terms to create the GCMF. This is foundational in factorizing polynomials, enabling simplification and solving.

Navigating Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations, representing a particular value or set of values. They may seem daunting, but breaking them into simpler parts, like finding the GCF or GCMF, makes understanding more approachable.
In our situation, each term in the polynomial \(6a^4 - 3a^3 + 9a^2\) is an algebraic expression itself, and collectively, they form a more complex expression.
Learning to deal with these involves practicing how to identify parts like coefficients and variable powers, making it easier to find common factors. It's akin to piecing together a puzzle; knowing how each part fits into the larger picture helps solve the overarching problem.
Key steps include:

• Recognizing and separating each term.
• Finding common factors among terms.
• Factorizing using these common elements.

Mastering these techniques begins with understanding simpler expressions before moving to more complex polynomials.