Problem 34 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 30 a^{2} b^{2}+40 a^{2} b^{2}+50 a^{2} b^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the given polynomial is \(120a^2b^2\).

Step 1: Identify the terms of the polynomial.

The polynomial has three terms. They are: 1. \(30a^2b^2\) 2. \(40a^2b^2\) 3. \(50a^2b^2\)

Step 2: Find the greatest common factor (GCF) of the terms.

We will find the GCF of the coefficients and the variables separately. For coefficients: The GCF of 30, 40, and 50 is 10. For variables: Both the terms have \(a^2\) and \(b^2\), so the GCF for variables is \(a^2b^2\). Combining both, the GCF of the terms is \(10a^2b^2\).

Step 3: Factor out the GCF.

Now, we will factor out the GCF from each term and write the polynomial as the product of the GCF and the remaining expression: $$ 10a^2b^2(3 + 4 + 5) $$

Step 4: Simplify the remaining expression.

Now, we will simplify the expression inside the parentheses: $$ 10a^2b^2(12) $$

Step 5: Write the final factored form.

Now we can rewrite the polynomial in its factored form: $$ 120a^2b^2 $$ So the factored form of the given polynomial is \(120a^2b^2\).

Key Concepts

Greatest Common Factor

When tackling the factoring of polynomials, the Greatest Common Factor (GCF) is pivotal. Think of the GCF as the largest factor that is shared among all terms within a polynomial. Finding this common factor helps simplify expressions and pull out parts of the expression that are common to each term. Let's break down how to find the GCF:

• **Identify Coefficients**: Look at the numerical parts of each term. For instance, the coefficients in the example are 30, 40, and 50. To find the GCF of these numbers, determine the largest number that divides all of them without leaving a remainder, which is 10 in this case.
• **Identify Variables**: Check the variable parts of each term. Here, every term has the variable expression \(a^2b^2\). Like the coefficients, this is common across all terms.

Together, they form the GCF: \(10a^2b^2\). By understanding and identifying the GCF, you're set up to simplify the original polynomial.

Monomials

A monomial is a fundamental part of polynomials, being nothing more than a single term. In the realm of algebra, a monomial consists of:

• **Numbers (Coefficients):** The numerical part of the monomial, e.g., 30 in \(30a^2b^2\).
• **Variables:** These represent unknown values, in the example provided they are \(a\) and \(b\).
• **Exponentiation:** Shows how many times a variable is multiplied by itself, such as the \(a^2\) and \(b^2\).

Understanding monomials is crucial as they are the building blocks of polynomials. So, each term (like \(30a^2b^2\)) with a single coefficient and variable expression is a monomial that can be simplified or factored when needed.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while retaining their original value. This process is vital in making complex expressions more manageable and understandable.Here's how you simplify:

• **Identify and Pull Out the GCF:** Start by factoring the greatest common factor from all terms. This makes the polynomial easier to work with, as seen when we factored out \(10a^2b^2\).
• **Combine Like Terms:** Once you factor out the GCF, simplify any remaining expression by combining like terms. For instance, in our example, we found \(10a^2b^2(3 + 4 + 5)\) and simplified it to \(10a^2b^2 \times 12\).

Simplifying expressions not only tidies up the expression itself but also makes the relationships between different components clearer. This skill is essential in both basic and complex algebraic calculations.