Problem 24 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ 7 x^{2} y^{3} z^{2}, \quad 7 x^{2} y^{3} z $$

Step-by-Step Solution

Answer

Answer: The other factor is z.

Step 1: Write down the given product and factor

The given product is \(7x^{2}y^{3}z^{2}\) and the given factor is \(7x^{2}y^{3}z\).

Step 2: Divide the product by the given factor

We can find the other factor by dividing the given product by the given factor. So we have: Other factor = \(\frac{7x^{2}y^{3}z^{2}}{7x^{2}y^{3}z}\)

Step 3: Simplify the expression

Now we have to simplify this expression by canceling out common terms in the numerator and the denominator: $$ \frac{7x^{2}y^{3}z^{2}}{7x^{2}y^{3}z} = \frac{7}{7} \cdot \frac{x^{2}}{x^{2}} \cdot \frac{y^{3}}{y^{3}} \cdot \frac{z^{2}}{z} $$

Step 4: Cancel out the common terms and find the other factor

Cancel out the common terms: $$ \frac{7}{7} \cdot \frac{x^{2}}{x^{2}} \cdot \frac{y^{3}}{y^{3}} \cdot \frac{z^{2}}{z} = 1 \cdot 1 \cdot 1 \cdot \frac{z^{2}}{z} $$ Now, we can find the other factor: $$ \frac{z^{2}}{z} = z $$ So, the other factor is \(z\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra and mathematics as a whole. They are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. For example, the expression \( 7x^2y^3z^2 \) is an algebraic expression because it employs numbers (7) and variables (\( x, y, \) and \( z \)) that are combined using multiplication.

Variables in algebraic expressions represent unknown values and can take on different numbers. They are crucial for solving equations, modeling real-world situations, and simplifying complex problems.

• Constants: Fixed numbers (like 7 in the example).
• Variables: Letters representing unknown numbers (such as \( x, y, \) and \( z \)).
• Terms: Parts of an expression separated by + or - signs (\( 7x^2y^3z^2 \) is a single term).
• Coefficients: Numbers multiplying the variables (7 is the coefficient of the term \( x^2y^3z^2 \)).

Understanding and simplifying these expressions is central to successful algebra practice.

Factoring

Factoring in algebra involves breaking down expressions into simpler parts, or factors, that can be multiplied together to obtain the original expression. The task of finding the other factor in this exercise is a perfect demonstration of factoring.

Given the expression \( 7x^2y^3z^2 \), this expression can be factored because it consists of products of variables. The process of factoring helps in simplifying expressions and solving equations. Here's how it works in this scenario:

• Identify all terms that share a variable or number.
• Divide them as individual elements, as shown with fractions in the solution.
• Cancel out like terms to simplify them.
• The remaining expression after simplification is the factor you need to find.

This exercise emphasizes how reducing an expression by finding its factors can make complex algebraic problems more manageable.

Division of Polynomials

The division of polynomials, such as dividing \( 7x^2y^3z^2 \) by \( 7x^2y^3z \), is key in arithmetic operations involving algebraic expressions. This division works similarly to dividing numbers but requires special attention to the variables and their exponents.

When dividing polynomials, the following steps can simplify the process:

• Write both the dividend (the polynomial to be divided) and the divisor.
• Divide the coefficients of corresponding terms.
• Subtract the exponents of similar variables by applying the rule \( x^a / x^b = x^{a-b} \). This applies to all variable terms, as shown in the solution where \( x^2 / x = x^{2-1} = x \).
• Cancel out terms that evaluate to 1 or simplify naturally.

Through this process, students learn to manage polynomial expressions efficiently, turn complex division problems into simple arithmetic, and develop a deeper understanding of how algebraic operations interact.