Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ \frac{6 a^{3} b^{3}-9 a^{2} b^{2}+3 a b^{4}}{3 a b^{2}} $$

Step-by-Step Solution

Answer

The polynomial expression is \(2a^2b - 3a + b^2\).

Step 1: Break Down the Numerator

The given expression in the numerator is \( 6a^3b^3 - 9a^2b^2 + 3ab^4 \). We need to look for a common factor in each term, which will help simplify the fraction later.

Step 2: Factor Out the Common Factor

Each term in the numerator shares a factor of \( 3ab^2 \). Factoring this out, we have:\[ 6a^3b^3 - 9a^2b^2 + 3ab^4 = 3ab^2(2a^2b - 3a + b^2) \].

Step 3: Cancel Common Factor in Numerator and Denominator

Now replace the numerator with its factored form: \[ \frac{3ab^2(2a^2b - 3a + b^2)}{3ab^2} \].Since \(3ab^2\) is a common factor in both the numerator and the denominator, it can be canceled out.

Step 4: Simplify the Expression

After canceling \(3ab^2\) from the numerator and the denominator, we are left with:\( 2a^2b - 3a + b^2 \). This is the simplified polynomial form of the initial given expression.

Key Concepts

Factoring Polynomials

Factoring polynomials involves writing a polynomial as a product of its smaller polynomial factors. It’s like breaking down a big problem into smaller, more manageable pieces. This can make simplifying expressions or solving equations much easier.

In our original exercise, we started with the polynomial: \(6a^3b^3 - 9a^2b^2 + 3ab^4\). The key to factoring is identifying the greatest common factor (GCF) from each term. Here, \(3ab^2\) was the GCF, as all terms share these elements.

To factor, we divide each term by \(3ab^2\) and write the polynomial as \(3ab^2(2a^2b - 3a + b^2)\). You now have the polynomial broken into simpler parts, which is easier to handle or further simplify. Understanding how to factor by finding common factors is a crucial skill that aids in many algebraic processes.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means reducing them to a form that's easier to understand or work with. This often involves factoring, canceling common terms, or combining like terms.

In our exercise, after factoring out the common term \(3ab^2\), we were left with \(3ab^2(2a^2b - 3a + b^2)\) in the numerator.

Next, because \(3ab^2\) was also in the denominator, we canceled it out. This left us solely with \(2a^2b - 3a + b^2\), a much simpler expression.

The process of simplifying can greatly decrease the complexity of solving equations or evaluating expressions, giving clearer insights and results.

Common Factors in Algebra

Common factors in algebra refer to a term or number that divides each component of a polynomial evenly. Identifying these factors is fundamental for simplifying expressions or solving equations.

In the exercise given, each component of the polynomial \(6a^3b^3 - 9a^2b^2 + 3ab^4\) shared a common factor of \(3ab^2\).

• For \(6a^3b^3\), factoring out \(3ab^2\) gives \(2a^2b\).
• For \(-9a^2b^2\), it leaves \(-3a\).
• For \(3ab^4\), factoring gives \(b^2\).

Recognizing and factoring out common factors allow us to reduce polynomial expressions to their simplest form, making further calculations straightforward. This method is a backbone operation in algebra, paving the way for more complex problem-solving.