Problem 55 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. \(15 a^{2} b^{2}-5 a b^{2} c^{2}\)

Step-by-Step Solution

Answer

The factored form is \(5ab^2(3a - c^2)\).

Step 1: Identify the Common Factors

Look for the greatest common factor (GCF) shared by the terms. In the expression \(15a^2b^2 - 5ab^2c^2\), the GCF is \(5ab^2\) since it is present in both terms.

Step 2: Factor Out the GCF

Divide each term by the GCF and factor it out. The expression \(15a^2b^2 - 5ab^2c^2\) becomes:\[5ab^2(3a - c^2).\]

Step 3: Check for Further Factorization

Check if the expression inside the parenthesis \(3a - c^2\) can be further factored. Since \(3a - c^2\) has no common factors or recognizable patterns, it is already in its simplest form.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a fundamental concept in factoring polynomials. It refers to the largest factor that is common to all terms within a given expression. Identifying the GCF is the first step when factoring any polynomial. This step is crucial because it simplifies the expression and makes further factorization steps more manageable.

To find the GCF, examine each term of the polynomial and break them down into their prime factors. For instance, in the polynomial expression \(15a^2b^2 - 5ab^2c^2\), identify common numeric and variable factors.

• The numeric GCF of 15 and 5 is 5.
• In terms of variables, both terms share \(ab^2\).

Thus, the GCF for the entire expression is \(5ab^2\). This factor is then divided out of each term in the expression to simplify it. Remember, finding the GCF sets the stage for further factorization steps, so it's a step you can't skip.

Polynomial Expressions

Polynomial expressions consist of terms that are composed of variables raised to whole-number exponents and coefficients. Understanding this basic structure is key to mastering other concepts like factorization. For instance, in our original expression \(15a^2b^2 - 5ab^2c^2\),

Note the following:

• The expression has two terms: \(15a^2b^2\) and \(-5ab^2c^2\).
• Each term includes variables and a numerical coefficient.
• The exponent indicates the degree of each variable within a term.

Understanding these components helps you apply factorization techniques effectively. Recognizing how terms are structured makes it easier to identify opportunities for simplification, such as common factors or patterns that allow for further factorization.

Factorization Techniques

Factorization techniques turn a complicated polynomial into simpler, more manageable expressions. Once you've identified and factored out the greatest common factor, you can explore other techniques.

Here’s the approach refined by our exercise:

• Firstly, factor out the GCF as efficiently as possible.
• Next, analyze the resulting expression to see if any further factorization is possible.

In our exercise, after extracting \(5ab^2\), we were left with \(3a - c^2\). We couldn't factor it further because there are no common factors in \(3a - c^2\) or apparent patterns like those seen in special products (e.g., perfect square trinomials or difference of squares). This end point – where no further factorization can simplify an expression – is crucial. It shows that the polynomial is reduced to its simplest form. Factorization isn't just about simplification; it's about honing your ability to see these structures and relationships within polynomial expressions.