Problem 21 - Technical Mathematics with Calculus

Question

$$8 x^{2} y^{2}+12 x^{2} z^{2}$$

Step-by-Step Solution

Answer

\(4x^{2}(2y^{2} + 3z^{2})\)

Step 1: Factor out the common term

First, identify the greatest common factor (GCF) in both terms. The GCF is the highest degree of common variables and constants that can be divided out from both terms. In this case, the GCF is \(4x^2\) because it is the largest factor common to both \(8x^2y^2\) and \(12x^2z^2\). Divide each term by \(4x^2\) and factor it out.

Step 2: Rewrite the expression

After factoring out the GCF \(4x^2\), rewrite the original expression as a product of the GCF and the remaining terms. This gives you the expression in its factored form.

Key Concepts

Greatest Common Factor

Understanding the concept of the greatest common factor (GCF) is vital in the process of simplifying algebraic expressions and especially when working with polynomials. The GCF is the highest number, or the most significant term, that divides exactly into two or more numbers or terms. For instance, when looking at the terms of a polynomial such as \(8x^2y^2\) and \(12x^2z^2\), we search for the largest numerical coefficient and highest power of any common variables.

In our example, both terms have the variable \(x^2\) and the numerical coefficients are 8 and 12. The GCF of the numerical coefficients is 4, because 4 is the largest number that divides both 8 and 12 without leaving a remainder. Similarly, since both terms contain \(x^2\), it is also part of the GCF. Combined, the GCF of the given terms is \(4x^2\).

Identifying the GCF is the first crucial step in factoring polynomials because it allows you to simplify and reduce the expression to its most elementary form. This simplification makes other types of factoring, such as splitting the middle term or factoring by grouping, considerably simpler.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations (such as addition, subtraction, multiplication, and division). The expression \(8x^2y^2 + 12x^2z^2\) is a typical example, where \(x\), \(y\), and \(z\) are variables that can take on various values.

One key aspect of algebraic expressions is that they can be manipulated through a variety of operations to simplify or solve them. The goal, particularly with a polynomial, is often to rewrite the expression in a form that reveals more information about its structure or to solve for a variable. This can involve combining like terms, using the distributive property, or factoring out common factors.

Terms and Coefficients

Each part of the expression separated by a plus or minus sign is called a 'term', and the numerical multiplier of the variables in a term is known as the 'coefficient'. In our example, \(8x^2y^2\) and \(12x^2z^2\) are terms where 8 and 12 are coefficients. Understanding the role of each component in an algebraic expression is fundamental to learning how to manipulate and factor these expressions successfully.

Polynomial Factoring Steps

Factoring polynomials is a crucial process in algebra that involves breaking down a polynomial into simpler components, or 'factors', that when multiplied together, give back the original polynomial. This process often simplifies the polynomial or prepares it for further algebraic operations.

The general steps for factoring polynomials typically include:

• Identifying the Greatest Common Factor: As with our example \(8x^2y^2+12x^2z^2\), the first step is always to determine if there is a common factor that can be factored out from all terms.
• Factoring out the GCF: Divide each term by the GCF and write the polynomial as a product of the GCF and the remaining polynomial.
• Look for Additional Factoring Opportunities: After removing the GCF, the polynomial may still be factorable. This could involve spotting patterns such as difference of squares, perfect square trinomials, or factoring by grouping if applicable.
• Checking Your Work: Multiply the factors to ensure they give the original polynomial, confirming the factoring process was correct.

It's important to proceed systematically and double-check each step to avoid errors. Recognizing patterns and practicing factoring with different types of polynomials increase proficiency in this skill, which is key in solving more advanced algebraic problems.